The Relativistic Spin Precession in the Compact Double Neutron Star System PSR J1946+2052

The discovery of pulsars in double neutron star (DNS) systems (e.g., PSR B1913+16 Hulse & Taylor 1975) unlocks ideal laboratories for testing theories of gravitation. Relativistic effects, such as Shapiro delay, Einstein delay, orbital period decay caused by gravitational-wave emission, and periastron advance, can be observed in pulsar timing through the measurements of the post-Keplerian parameters of pulsar binary systems (Damour & Deruelle 1985, 1986). The timing of PSR B1913+16 and PSR J0737-3039A/B has already stringently tested general relativity (GR), for example, by Weisberg & Taylor (2005), Kramer et al. (2006, 2021), and Weisberg & Huang (2016).

Relativistic spin precession is another relativistic effect caused by the curvature of spacetime in the presence of a large mass object. This precession causes the pulsar's spin vector to precess around the total angular momentum vector and moves the pulsar's emission beam with respect to our line of sight. The relativistic spin precession is usually not directly detectable through pulsar timing, but sometimes can be observed through the temporal evolution of the pulse profile and its polarization.

The first indication for profile changes caused by relativistic spin precession was given by Weisberg et al. (1989), observing the relative amplitude of the two prominent components of PSR B1913+16's profile changing over time. A systematic decrease in the pulse width change was later detected, and the profile changes were analyzed and modeled by Kramer (1998). A quantitative test of the precession rate was not possible, but the changes were found to be consistent with the expectation from GR. After PSR B1913+16, the relativistic spin precession has been detected through profile change in more pulsar binary systems, such as PSR J1537+1155 (Stairs et al. 2004), PSR J1141-6545 (Hotan et al. 2005), PSR J0737-3039B (Burgay et al. 2005), PSR J1906+0746 (Lorimer et al. 2006; Desvignes et al. 2019), and more recently, in PSR J1757-1854 (Cameron et al. 2023). Some pulsars in DNSs, such as PSR J0737-3039A, show no sign of temporal profile evolution. This could be due to a small misalignment angle between the pulsar's spin vector and the orbital angular momentum, as inferred for PSR J0737-3039A (Ferdman et al. 2013). As the swing in the polarization position angle (PPA) is a marker of pulsar geometry, the temporal evolution of the PPA is also a potential (and powerful) indicator of relativistic spin precession (Stairs et al. 2004; Desvignes et al. 2019).

Assuming GR, the rate of spin precession of the pulsar is determined from the system's Keplerian parameters and two objects' masses (Barker & O'Connell 1975):

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{\mathrm{SO}}=\displaystyle \frac{1}{2}{\left(\displaystyle \frac{{{GM}}_{\odot }}{{c}^{3}}\right)}^{2/3}{\left(\displaystyle \frac{{P}_{b}}{2\pi }\right)}^{-5/3}\displaystyle \frac{{m}_{2}(4{m}_{1}+3{m}_{2})}{(1-{e}^{2}){\left({m}_{1}+{m}_{2}\right)}^{4/3}},\end{eqnarray} \tag{ 1 }$$

where G is the gravitational constant, c is the velocity of light, P_b is the orbital period, e is the orbital eccentricity, m₁ and m₂ are masses of the pulsar and companion measured in units of solar mass. The Keplerian parameters such as P_b and e can be derived through pulsar timing, and masses can be measured if more than two post-Keplerian parameters are detected. If the spin precession rate is derived independently through the profile and PPA evolution, then it can be used to test GR (Stairs et al. 2004; Fonseca et al. 2014; Desvignes et al. 2019). In the case of PSR J1906+0746, the measured spin precession rate agrees with GR to within 5% (Desvignes et al. 2019). In the particular case of PSR J0737-3039A/B, the spin precession rate is determined through modeling pulsar B's eclipse morphology evolution, and that leads to a rate in agreement with GR to within 13% in Breton et al. (2008) and recently 6% in Lower et al. (2024). At last, if the spin precession rate could not be directly measured, one still can use the relativistic spin precession to model the binary system's geometry by assuming GR is correct (Kramer 1998; Manchester et al. 2010; Ferdman et al. 2013).

Stovall et al. (2018) discovered the DNS system PSR J1946+2052, which has the shortest orbital period among known DNS systems at 1.88 hr. The eccentricity of the orbit is 0.064. They detected only one post-Keplerian parameter, the periastron advance, to be 256 yr⁻¹, indicating the system's total mass is 2.50(4) M_⊙ under the validity of GR. The pulsar was considered to be the primary neutron star of the system, due to the spin period of 16.9 ms and the magnetic field strength at the surface of 4 × 10⁹ G. They suggested that this system resembles the double pulsar system PSR J0737-3039A/B in configuration and binary evolution. Therefore, they proposed that the pulsar's spin vector could be nearly aligned with the orbital angular momentum vector and thus spin precession should be hard to observe. In their work, they did not detect any polarization.

The Five-hundred-meter Aperture Spherical radio Telescope (FAST) is a single-dish radio telescope (Nan 2008; Nan et al. 2011) and has been operating since 2019 (Jiang et al. 2020). Owing to the high sensitivity of FAST, here we report the detection of polarization and significant profile changes for PSR J1946+2052 for the first time. In Section 2, we describe the data reduction and specifically demonstrate the removal of the ionospheric Faraday rotation effect. In Section 3, we show the results of our analysis of the pulse profile and polarization and model the geometry of this pulsar system. In Section 4, we discuss the evolution of this DNS and predict how the measurement of spin precession rate would advance in future observations.

We observed PSR J1946+2052 with the center beam of FAST's 19 beam receiver at a central frequency of 1250 MHz using a bandwidth of 500 MHz for 11 epochs. We display the information of the pulse profile, polarization and PPA of our observations in Figure 1. The first session, conducted on 2019 March 30, was observed in the two-polarization (AA, BB) mode, while the remaining sessions were carried out in full Stokes mode in 2021 and 2022. Details of the length and date of observations are shown in Table 1. We recorded all data in PSRFITS (Hanisch et al. 2001) format with a sampling time of 49.152 μs and 8 bit digitization. To calibrate each observation's polarization, we pointed the telescope off the pulsar and turned on the noise diode in a 0.2 s cycle for 1 minute before observing the pulsar.

We used the DSPSR (van Straten & Bailes 2011) analysis program and the PSRCHIVE (Hotan et al. 2004) software package to fold and de-disperse the pulsar data, to eliminate radio frequency interference, and to calibrate the polarization. We obtained the ephemeris from PSRCAT ¹¹ (Manchester et al. 2005) and updated it with our own time-of-arrival measurements using TEMPO ¹² (Nice et al. 2015). With the local solution of the ephemeris derived from TEMPO, for each epoch, we folded the data and scrunched time. To avoid effects due to uncertainties in the long-term ephemeris, we aligned pulses from different observations by using the maximum of the normalized pulse profile. The rotation measure (RM) was measured using rmfit in PSRCHIVE. However, due to the low signal-to-noise ratio (S/N) and low degree of linear polarization, we could not derive a precise RM for the whole set of observations. Instead, we used the observation with the highest S/N to obtain RM = –27(11) rad m⁻² and removed the Faraday rotation effect in polarization with a fixed RM −27 rad m⁻² to analyze the relative change of the geometrical parameters (see Section 3). Then, we computed the Stokes parameters I, Q, U, and V and exported the archive data into a NumPy (Perez & Granger 2007) readable format using the Python package of PSRCHIVE.

RM is a crucial parameter in studying pulsar polarization. For example, 1 rad m⁻² shifts the PPA by almost 3° at 1250 MHz. As explained in Sotomayor-Beltran et al. (2013),

$$\begin{eqnarray}&&\mathrm{RM}={\mathrm{RM}}_{\mathrm{int}}+{\mathrm{RM}}_{\mathrm{ion}}+{\mathrm{RM}}_{\mathrm{ism}}+{\mathrm{RM}}_{\mathrm{igm}},\end{eqnarray} \tag{ 2 }$$

where RM_int, RM_ion, RM_ism, and RM_igm denote the intrinsic RM and RM of the ionosphere, interstellar medium, and intergalactic medium respectively, where the latter is not applicable in this case. We can also expect a change in RM_ism to be small, based on typical lines of sight with a relatively uniform magnetic field along those and conservative estimates of a δDM of approximately 10⁻⁴ pc cm⁻³. Hence, we consider mainly the influence of ionospheric RM (RM_ion) and its variation on the measurement of the PPA.

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We calculated the RM_ion of each epoch by using the ionFR package (Sotomayor-Beltran et al. 2013) and global ionospheric map (GIM) products. The GIM products were generated by different analysis centers using total electron content gathered from global positioning system stations around the world, combined with a specific mathematical model (Li et al. 2017). For each observation, dozens of GIM analysis centers can provide measurements of the ionospheric RM, and their estimates may differ slightly, due to model dependence. Figure 2 shows an example of the RM_ion of PSR J1946+2052 derived from GIM products on 2021 March 9, which are available on NASA's website. ¹³ To remove the bias in the GIM products, we interpolated values of RM_ion and their errors during the time span of our observations and obtained the weighted average and error for each analysis center. Then, we weighted these RM_ions from different analysis centers with their errors and derived the average and the uncertainty of each day's ionospheric RM. The RM_ion values for each observation epoch are presented in Table 1.

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We have detected polarization in observations with FAST lasting longer than 1 hr. The profiles at 1.25 GHz are presented in Figure 1 with 1024 phase bins. We consider the emission feature near pulse phases of ∼0.36–0.40 to be the interpulse, complementing the stronger main pulse at pulse phases of ∼0.77–0.93. The observation in 2019 has no polarization information so we show only total intensity (black line) for this epoch in Figure 1. For the other epochs, we show linear polarization (red line), circular polarization (blue line), and the measured PPAs in the upper part of the plots. To obtain the uncertainty in the PPAs, we consider the unbiased linear polarization L_true, as detailed in Everett & Weisberg (2001). As scintillation modifies the S/N from epoch to epoch, the number of well-defined PPA points varies accordingly. The shown PPA values are corrected for Faraday rotation, due to the ionospheric RM_ion contribution by subtracting λ² × RM_ion from the PPA values (measured in radians), where λ is the observing wavelength.

As shown in Figure 1, we have observed significant changes in the pulse profiles of PSR J1946+2052 over a 3 yr time span, which can be attributed to relativistic spin precession. The changes manifest themselves most prominently in a change in the relative amplitudes of both interpulse and main pulse over time. To obtain a smooth pulse profile with less noise, we downsample the spin phase to 256 bins to extract the double-peak structure of both the interpulse and main pulse. We normalize each integrated profile to the peak of the main pulse's leading component, and the relative amplitudes of other components are shown in panels (b) and (c) of Figure 3. The uncertainties in the relative amplitudes are calculated as the rms of each profile's off-pulse region.

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Relativistic spin precession can cause changes in the pulse profile and the PPAs. To reveal these effects, we analyze the full polarization profile of PSR J1946+2052. We use a simple circular hollow-cone-shaped emission beam to model the profile change. For polarization, the swings in the PPA of PSR J1946+2052 are not a perfect S-shape, and the points shown in gray are not easily consistent with a typical swing in the rotating-vector model (RVM). They may be caused by an unresolved orthogonal jump, as their location coincides with a minimum in linear polarization and a change in the sign of circular polarization. In our analysis, we then ignore this small range of PA values (displayed as gray points in Figure 1). We present details of our analysis in the following sections.

3.1. Pulse Profile and PPA Evolution

The most important profile evolution is that the interpulse has only one peak on 2019 March 30 but then becomes double-peaked in observations taken after 2021 January 13. Assuming a circular hollow-cone-shaped emission beam, this profile variation can be modeled by the movement of our line of sight from the edge of the emission beam to the interior of the hollow cone. This suggests that initially, the line of sight was near the edge of the beam, but our assumption of the beam shape may be wrong (see Desvignes et al. 2019).

To estimate the separations between the two components of the interpulse and the main pulse, we use two Gaussian functions with identical widths to fit these two pulse components separately in each observation. The separation between the components is determined as the distance between the centers of the two Gaussians. The separation between the interpulse and the main pulse does not enter the analysis at this stage. For the interpulse seen on 2019 March 30 with only one peak, this method cannot be applied. Here, we fix the widths of the two Gaussian functions to the average value obtained from later observations. This fixed width is consistent with the distribution of the widths we derive from the 2021–2022 epochs. We also make the two Gaussian functions have the same amplitude. With these additional constraints, we can obtain an estimated peak component separation for the 2019 observation. We display the Gaussian functions that are used to fit the interpulse and the main pulse in panel (a) of Figure 3. The peak component separations obtained using these methods are shown in panels (d) and (e) of Figure 3.

Figure 4 displays the stacked PPA measurements obtained from PSR J1946+2052, which shows no clear evidence of evolution. The relatively large uncertainties and the absence of significant changes in the shape of the swing suggest that the expected spin precession has not yet reached a level of detectability through PPA measurements. This may be caused by the geometrical configuration, a special precession phase, or oversized uncertainties in our measurements.

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3.2. Geometry Modeling

3.2.1. Convention and Definitions

In this study, we adopt the PSR/IEEE convention (van Straten et al. 2010) to define the geometrical angles and their relationships, which is commonly referred to as the observer's convention. These angles are illustrated in Figure 5. It should be noted that this convention differs from the convention used in previous studies, such as those of Damour & Taylor (1992) and Kramer & Wex (2009) (hereafter referred to as the DT convention), in which the line of sight is defined as pointing out from the sky plane. The angle between the line of sight and the spin vector (λ) and the inclination angle of the orbit (i) are complementary in these two conventions. Apart from those changes in convention, the formalism below follows that of Kramer & Wex (2009).

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As the spin angular momentum of the pulsar is much smaller than its total angular momentum, we can assume that the direction of the orbital angular momentum is constant with time and is the same as the total angular momentum. This means that the inclination angle i remains constant with time. As the pulsar's spin vector precesses around the orbital angular momentum, the line of sight will cut through different latitudes of the emission beam, causing the angle λ to change with time according to the following function:

$$\begin{eqnarray}&&\cos \,\lambda (t)=\cos \,\delta \,\cos \,i+\sin \,\delta \,\sin \,i\,\cos \,{{\rm{\Phi }}}_{\mathrm{SO}}(t),\end{eqnarray} \tag{ 3 }$$

where δ is the misalignment angle of the spin vector from the orbital angular momentum vector. Φ_SO(t) is the precession phase and is described as

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{\mathrm{SO}}(t)={{\rm{\Phi }}}_{0}+{{\rm{\Omega }}}_{\mathrm{SO}}(t-{T}_{0}),\end{eqnarray} \tag{ 4 }$$

where Φ₀ is the reference precession phase, T₀ is the time when the precession phase is Φ₀, and Ω_SO is the rate of the precession. In our work, we define T₀ as the date of the first observation (2019 March 30).

The projected position angle of the spin vector (Ψ₀) on the celestial sphere changes, due to spin precession. In our convention, the position angle equals 0 in the north of the sky plane ( I ₀) and increases counterclockwise as turning to the east ( J ₀). Ψ₀ is described as

$$\begin{eqnarray}&&{{\rm{\Psi }}}_{0}(t)={{\rm{\Omega }}}_{\mathrm{asc}}+{\rm{\Delta }}{\psi }_{{\rm{A}}}+{{\rm{\Delta }}}_{{\rm{F}}}+\eta (t),\end{eqnarray} \tag{ 5 }$$

where Ω_asc is the ascending node of the orbital plane, Δ_A is the angle shift caused by aberration, Δ_F is the change of the Faraday rotation effect, which is illustrated in Equation (2), and η(t) is the longitude of the pulsar's spin precession. Because it is difficult to distinguish the specific values of Ω_asc, Δψ_A, and Δ_F, we simply define Ψ₀(t) as

$$\begin{eqnarray}&&{{\rm{\Psi }}}_{0}(t)={\rm{\Delta }}{\rm{\Psi }}+\eta (t),\end{eqnarray} \tag{ 6 }$$

and η(t) can be derived from

$$\begin{eqnarray}&&\cos \,\eta (t)=\displaystyle \frac{\sin \,\delta \,\sin \,{{\rm{\Phi }}}_{\mathrm{SO}}(t)}{\sin \,\lambda (t)},\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&\sin \,\eta (t)=\displaystyle \frac{\cos \,\lambda (t)\cos \,i-\cos \,\delta }{\sin \,i\,\sin \,\lambda (t)}.\end{eqnarray} \tag{ 8 }$$

3.2.2. Modeling Pulse Profile Separations and PPAs

We can model the separation evolution by assuming a circular hollow-cone-shaped emission beam. For the dipole field emission, the separation between the interpulse and the main pulse could deviate from 180° due to the different emission heights, while the shape of these two pulses could be the hollow-cone shape. We assume each hollow-cone-shaped beam to be symmetrical so that the separation between two peak components in one pulse can be expressed as (Kramer 1998)

$$\begin{eqnarray}&&{\sin }^{2}\left(\displaystyle \frac{W(t)}{4}\right)=\displaystyle \frac{{\sin }^{2}(\rho /2)-\sin {\,}^{2}(\beta (t)/2)}{\sin \,\alpha \,\sin (\alpha +\beta (t))},\end{eqnarray} \tag{ 9 }$$

where W(t) is the separation, ρ is the opening angle of the emission beam, α is the magnetic inclination angle, and β(t) is the angle between the magnetic axis and the line of sight. β(t) is given by

$$\begin{eqnarray}&&\beta (t)=\lambda (t)-\alpha .\end{eqnarray} \tag{ 10 }$$

Once we determine α for the main pulse, the interpulse's magnetic inclination angle is given by π − α. We have observed the interpulse changing from a one-peak to a two-peak shape, but the short time span covered by our observations alone is insufficient to constrain the six geometrical parameters that we need to model the precession.

For modeling the PPAs, we continue to adopt the precessional RVM (Kramer & Wex 2009) that considers relativistic spin precession. The conventional RVM is used to model the PPA that increases clockwise. However, the PPA we use increases counterclockwise as we derive the Stokes parameters from PSRCHIVE, which follows the PSR/IEEE convention (van Straten et al. 2010). To modify the RVM equation accordingly, we refer to Everett & Weisberg (2001). As Ψ₀(t) and β(t) vary with time due to precession, the precessional RVM can be described by Equation (11) as

$$\begin{eqnarray}\begin{array}{rcl} & & {\rm{\Psi }}(t)={{\rm{\Psi }}}_{0}(t)\\ & & +{\tan }^{-1}\left(\displaystyle \frac{\sin \,\alpha \,\sin ({\phi }_{0}-\phi )}{\sin \,\lambda (t)\,\cos \,\alpha -\cos \,\lambda (t)\,\sin \,\alpha \,\cos ({\phi }_{0}-\phi )}\right),\end{array}\end{eqnarray} \tag{ 11 }$$

where ϕ is the spin phase, ϕ₀ is the reference phase of the spin axis and is relative to the main pulse. Due to the limited S/N of our polarization data, we cannot consider the difference in emission height between the interpulse and the main pulse through PPAs, as done in Johnston & Kramer (2019). Although the reference phase of the spin axis position angle is not expected to change, due to relativistic spin precession, we fit ϕ₀ for every epoch because the profiles are aligned based on the maximum of the main pulse. We exclude the part of the PPA where apparent orthogonal polarization modes are present. The selected PPA points are shown in Figure 1 as black points.

In order to constrain the geometry using profile separations and PPAs, we combine the profile separation model and precessional RVM and reconstruct the likelihood as given by Equation (12).

$$\begin{eqnarray}&&\begin{array}{l}\mathrm{ln}{ \mathcal L }=-\displaystyle \frac{1}{2}\displaystyle \sum _{i=1}^{{N}_{\mathrm{MJD}}}\left(\displaystyle \sum _{j=1}^{{N}_{\mathrm{PPA},i}}{C}_{\mathrm{PPA},i}{\left(\displaystyle \frac{{{\rm{\Psi }}}^{\mathrm{obs}}-{\rm{\Psi }}}{{\sigma }_{\mathrm{PPA}}^{\mathrm{obs}}}\right)}_{j}^{2}\right.\\ \,+\,{C}_{\mathrm{inter}}{\left(\displaystyle \frac{{W}_{\mathrm{inter}}^{\mathrm{obs}}-{W}_{\mathrm{inter}}}{{\sigma }_{\mathrm{inter}}^{\mathrm{obs}}}\right)}^{2}\\ \,{\left.+\,{C}_{\mathrm{main}}{\left(\displaystyle \frac{{W}_{\mathrm{main}}^{\mathrm{obs}}-{W}_{\mathrm{main}}}{{\sigma }_{\mathrm{main}}^{\mathrm{obs}}}\right)}^{2}\right)}_{i},\end{array}\end{eqnarray} \tag{ 12 }$$

where i and j indicate each day and each PPA point in day i's observation. As shown in Equation (12), we weigh each component of the likelihood with the number of data points as follows:

$$\begin{eqnarray}&&{C}_{\mathrm{inter}}={C}_{\mathrm{main}}=1,{C}_{\mathrm{PPA},i}=\displaystyle \frac{{N}_{\mathrm{inter}}}{{N}_{\mathrm{PPA},i}},\end{eqnarray} \tag{ 13 }$$

where C_inter, C_main, and C_PPA,i are the weighted coefficients we used for the component separations of the interpulse, the main pulse, and each day's swing in the PPA. N_inter is the number of data points of the interpulse's separation. Note that N_inter = N_main because we obtain one data point for the component separation of the main pulse and interpulse in each observation. N_PPA,i indicates the number of data points of each day's swing in the PPA.

This likelihood enables us to fit the profile separations with RVM information of β and ensures that β < ρ. We utilize the Markov Chain Monte Carlo (MCMC) method based on the Python packages emcee (Foreman-Mackey et al. 2013) and corner (Foreman-Mackey 2016) to obtain the distribution of geometrical parameters. Since we have no prior information on the geometry, we search for the optimal geometry parameters with uniform priors over the allowed range while assuming that GR is correct. The individual masses of the binary system are not accurately determined yet. Still, we can use the mass function (0.268184(12) M_⊙) and total mass (2.50(4) M_⊙) of this system from Stovall et al. (2018) to express the precession rate as a function of the orbital inclination angle i.

We display the MCMC result of geometrical angles in Figure 6 and the ϕ₀ of each epoch in Figure 7. We present in Figure 4 the best fit of the swings in the PPA, while the best fit of separations and 1σ samples are displayed in Figure 3. Our results show that δ is very small, and is consistent with the expectation by Stovall et al. (2018). The value of α is related to the main pulse and indicates that the pulsar is a nearly orthogonal rotator. The observed significant difference in profile separations between the interpulse and the main pulse is because the spin axis and the magnetic axis are not entirely orthogonal. Given the small value of δ, we cannot derive a precise precession phase Φ₀. Assuming negligible effects from additional Faraday rotation and aberration, the longitude of the ascending node Ω can be determined to a small range. The inclination angle i has been constrained by the fit to ${i={63}^{\circ }}_{-{3}^{\circ }}^{+{5}^{\circ }}$, which is consistent with the updated timing result (P. C. C. Freire, private communication), where i is estimated to be between 65° and 80°. Based on our estimate of i, we can predict the companion mass and the precession rate to be 1.33(4) M_⊙ and 7.96(23)° yr⁻¹, respectively. Thus, the mass of PSR J1946+2052 is predicted to be 1.17(6) M_⊙. The distribution of these two parameters is shown in Figure 8. By calculating β with the best-fit geometrical parameters (the dark red curve in Figure 9), we predict that the separation of the interpulse will disappear in 2032 because our line of sight will move out of the hollow-cone-shaped emission beam. Due to the small amplitude of β, the separation of the main pulse will stay visible all the time. We display the evolution of β in Figure 9. With improved data and longer time baselines, in the future, we may be able to combine the timing analysis with profile and polarization studies to derive the geometry of this system precisely and even test GR.

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Through our analysis of the profile separations and PPAs, we are able to determine that the spin–orbit misalignment angle δ of the PSR J1946+2052 system is very small, approximately 02 (as shown in Figure 6). Due to the pulsar's short spin period, it appears to be a recycled pulsar and therefore the primary neutron star in the system. The small value of δ suggests the spin axis and the orbital angular momentum to be nearly aligned, consistent with the hypothesis that the kick from the second supernova (SN) was small (Stovall et al. 2018). It is worth noting that another system with a small δ (<32) is the recycled pulsar in the double pulsar system, PSR J0737-3039A (Ferdman et al. 2013). The small δ and low eccentricity of both PSR J0737-3039A (e = 0.088) and PSR J1946+2052 (e = 0.064) imply that the second SN in each system was likely low-kick and symmetric, with little mass loss. In the case of the double pulsar, accurate mass measurements of both pulsars through pulsar timing have provided additional insight into the nature of the second SN. The low mass of the companion pulsar PSR J0737-3039B and the low system velocity suggests that the second SN in this system could be an ultrastripped SN (Tauris et al. 2013, 2017) or an electron-capture SN (Ferdman et al. 2013). Based on the companion mass we derive, we infer that the second SN of the PSR J1946+2052 system is likely similar to that of the double pulsar system. Ferdman et al. (2013) did not find obvious profile evolution of PSR J0737-3039A from 2005–2011, confirmed more recently by Hu et al. (2022). The reason why we nevertheless observe profile evolution in PSR J1946+2052 is likely that our line of sight is closer to the edge of the circular emission beam than that in PSR J0737-3039A; thus, the variation of the peak separation is significant.

The trend of the separation change is different for the interpulse and the main pulse. The separation of the interpulse is increasing, while that of the main pulse appears to be decreasing. The variation in the interpulse from one peak to two peaks suggests that our line of sight is moving toward the magnetic axis of the interpulse if the beam is a hollow cone. In this work, we make the assumption that the two radio beams of the pulsar are symmetrical. Based on our fitting result, the line of sight is near the edge of two emission beams and moving toward the center of the beams. Therefore, we would expect the separation between the main pulse peaks to increase as well, which is, however, the opposite. This could be due to the complexity of the emission pattern in the interior of the main pulse's emission beam, as shown in PSR J1906+0746 (Desvignes et al. 2019).

In this work, we have assumed the validity of GR and used a simple beam model to constrain binary parameters. To determine when we can utilize PSR J1946+2052 for testing GR with spin precession, we conduct simulations of future observations of profile separations and the swing in the PPA, assuming that our current best-estimated geometry is accurate. We produce a set of simulated profile peak separations and swings in the PPA every 2 months, based on the projections from our best-fit parameters, and incorporate the precession rate we derive from these parameters. To account for measurement errors on the actual profile peak separations and swings in the position angle, we introduce noise to our projected data. The error of each data point is determined by using the bootstrap method among corresponding existing errors. As the number of new simulated swings in the PPA grows faster than the number of simulated profile component separations, the swings in the PPA will dominate the global fit. To counterbalance this discrepancy, we resample the simulated swings in the PPA and average the values for each year. This should not impact our fit outcome, as the position angle swing varies very slowly with time.

We perform simulations for additional 3, 6, and 9 yr data sets, and present the results in Table 2, except for ϕ₀ of each epoch. We fit for the value of Ω_SO without assuming GR and find that it can be constrained to a relative precision of only 43% in 9 yr (total of 11 yr). However, these simulations do not incorporate pulsar timing. With 11 yr FAST timing, we could possibly derive the DNS's masses and inclination angle i as well as potentially the system's longitude angle of the ascending node Ω_asc. The timing measurement could be used as the prior in our analysis of the profile and PPA, significantly improving the constraint on the other geometrical parameters and the spin precession rate under the assumption of GR. This improvement is important for the pulsar emission model and binary evolution.

Table 2. Geometrical Parameters in Simulations

Data	α	ΔΨ	δ	Φ0	i	ρinter	ρmain	ΩSO
	(deg)	(deg)	(deg)	(deg)	(deg)	(deg)	(deg)	(deg yr−1)
3 yr	${86.81}_{-0.29}^{+0.28}$	110 ± 1	${0.35}_{-0.24}^{+0.85}$	${144}_{-31}^{+17}$	66 ± 1	27 ± 1	24 ± 1	${{\bf{6}}}_{-3}^{+5}$
6 yr	86.78 ± 0.15	108 ± 1	${0.45}_{-0.31}^{+0.83}$	${150}_{-26}^{+12}$	65 ± 1	27.78 ± 0.45	24.53 ± 0.29	${{\bf{5}}}_{-2}^{+5}$
9 yr	${86.68}_{-0.11}^{+0.10}$	112 ± 1	${0.30}_{-0.15}^{+0.38}$	${142}_{-18}^{+13}$	${66.43}_{-0.45}^{+0.39}$	26.64 ± 0.40	${23.39}_{-0.27}^{+0.26}$	${{\bf{7}}}_{-2}^{+3}$

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In summary, we observed PSR J1946+2052 11 times and found significant profile evolution and attributed this evolution to relativistic spin precession. We use the circular hollow-cone-shaped emission beam to derive the geometry of this system. We derive a global fit model by combining the separation fit and the PPA fit to estimate the geometrical angles. In the fit, we follow the results of PSR J1946+2052's mass function and total mass in Stovall et al. (2018) and assume GR to be correct. The result shows that the spin vector of this pulsar is nearly aligned with the orbital angular momentum, indicating that the second SN should be fast and symmetric, as suggested by Stovall et al. (2018). We also constrain the companion mass and the precession rate under the scenario of GR. We emphasize that the derived uncertainties do not take into account possible systematic shortcomings of the assumed beam model. The future timing of PSR J1946+2052 will improve our understanding of the pulsar emission model as well as binary evolution and constrain the spin precession rate under the assumption of GR.

We appreciate Emmanuel Fonseca for the helpful comments and discussions. This work was supported by the National SKA Program of China (2020SKA0120200, 2020SKA0120300), the National Natural Science Foundation of China (12203072, 12041303, 12203070), and the CAS-MPG LEGACY project. L.S. is supported by the Max Planck Partner Group Program funded by the Max Planck Society. X.M. and M.X. are supported by the Cultivation Project for FAST Scientific Payoff and Research Achievement of CAMS-CAS. J.Y. is supported by the Natural Science Foundation of Xinjiang Uygur Autonomous Region (grant No. 2022D01D85) and the Major Science and Technology Program of Xinjiang Uygur Au320.50tonomous Region (grant No. 2022A03013-2).