2.1 Evolution of Isolated Pulsars

Isolated pulsars are assumed to spin down via magnetic braking with braking index $n=3$ . Their surface magnetic fields may decay over time, though the decay timescale remains uncertain. Field decay is thought to arise from the Hall effect, which redistributes magnetic energy and enhances Ohmic dissipation (Romani Reference Romani1990; Konar & Bhattacharya Reference Konar and Bhattacharya1997). In this work, we adopt a simple exponential decay model, consistent with previous studies (Kiel et al. Reference Kiel, Hurley and Bailes2008; Chattopadhyay et al. Reference Chattopadhyay, Stevenson and Hurley2020; Dirson et al. Reference Dirson, Pétri and Mitra2022; Song et al. Reference Song, Stevenson and Chattopadhyay2024):

(1) \begin{equation} B_f = (B_i - B_{\textrm{min}}) \exp(-t / \tau) + B_{\textrm{min}},\end{equation}

where $B_i$ and $B_f$ are the initial and final magnetic fields, $B_{\textrm{min}} = 10^{8}$ G (Zhang & Kojima Reference Zhang and Kojima2006), and $\tau$ is the decay timescale.

The spin period derivative, assuming vacuum dipole radiation, is

(2) \begin{equation} \dot{P} = \frac{8 \pi ^ 2 R ^ 6 B^2 \sin^2{\alpha}}{3 c^3 I P} , \end{equation}

where $R=10$ km, c is the speed of light, $\alpha$ is the inclination angle, and the NS moment of inertia is (Lattimer & Schutz Reference Lattimer and Schutz2005)

(3) \begin{equation} I = (0.237 \pm 0.008) M R^2 \left[1 + 4.2\frac{\textrm{M}}{\textrm{M}_{\odot}} \frac{\textrm{km}}{\textrm{R}} + 90\left(\frac{\textrm{M}}{\textrm{M}_{\odot}}\frac{\textrm{km}}{\textrm{R}}\right)^4\right] , \end{equation}

Integrating Eq. 2 with B(t) from Eq. 1 gives

(4) \begin{equation} P^2 = \frac{16 \pi ^ 2 R ^ 6 \sin^2{\alpha}}{3 c^3 I} \left[B_{\textrm{min}}^2 t - \tau B_{\textrm{min}} (B_f - B_i) - \frac{\tau}{2}(B^2_f - B^2_i)\right] + P_0^2 . \end{equation}

where $P_0$ is the initial spin period. The evolution of P depends on $\alpha$ . Although $\alpha$ may evolve with B (Dirson et al. Reference Dirson, Pétri and Mitra2022), here we assume a fixed $\alpha$ , decoupling its evolution from P and B. Observations suggest older pulsars align their axes (Young et al. Reference Young, Chan and Burman2010), but modeling of radio and pulsars favors large $\alpha$ values (Johnston et al. Reference Johnston, Smith and Karastergiou2020). Accordingly, we set $\alpha = \pi/2$ .

2.2 Evolution of Pulsars in Roche Lobe Overflow

Roche lobe overflow (RLOF) occurs when a star fills its Roche lobe and transfers mass through the inner Lagrangian point, often forming an accretion disk. The Roche lobe radius is approximated by Eggleton (Reference Eggleton1983):

(5) \begin{equation} \frac{ R_{\rm L}}{a} = \frac{0.49q^{2/3}}{0.6q^{2/3} + \ln{(1+q^{1/3})}}, \end{equation}

where $R_{\rm L}$ is the Roche-lobe radius, a the orbital separation, and $q = M_{\rm comp}/M_{\rm NS}$ the companion-to-NS mass ratio.

Accretion during RLOF drives (i) magnetic field decay through burial and (ii) NS spin-up. Field decay follows

(6) \begin{equation} B = (B_0 - B_{\textrm{min}}) \exp(-\Delta M / \Delta M_d) + B_{\textrm{min}},\end{equation}

where $\Delta M$ is the accreted mass, $\Delta M_d$ a characteristic mass scale, $B_0$ the initial field, and $B_{\rm min}$ the minimum allowed field. The rate of angular momentum increase during accretion is

(7) \begin{equation} \dot{J}_{\boldsymbol{\rm{acc}}} = \varepsilon V_{\textrm{diff}} R^2_A \dot{M}_{\textrm{NS}}, \end{equation}

where $\varepsilon=1.0$ is the efficiency factor, $\dot{M}_{\textrm{NS}}$ the accretion rate, $V_{\textrm{diff}}$ the difference between the Keplerian angular velocity at the magnetic radius and the co-rotation velocity, and $R_A$ half the Alfvén radius:

(8) \begin{equation} R_{\rm Alfven} = \left(\frac{8 R^{12} B^4}{M \dot{M}^2 G}\right)^{\frac{1}{7}}.\end{equation}

The final spin period after mass transfer is

(9) \begin{equation} P_{\rm f} = 2 \pi / (2 \pi/P_{\rm i} + \Delta J_{\rm acc} / I ),\end{equation}

where $P_{\rm i}$ is the initial spin period, $\Delta J_{\rm acc}$ the angular momentum gained, and I the moment of inertia (Eq. 3). The corresponding $\dot{P}$ is computed by following Song et al. (Reference Song, Stevenson and Chattopadhyay2024).

2.3 Evolution of Pulsars in Common Envelope

Common envelope (CE) evolution occurs when the donor engulfs its companion in a shared envelope, typically leading to dramatic orbital shrinkage. CE evolution is critical for forming tight binaries that may merge as gravitational-wave sources. NS behavior during CE is uncertain, so we implement three scenarios.

1. (1) No accretion during CE; NS spins down naturally.

2. (2) NS accretes via an accretion disk, as in RLOF, allowing spin-up.

3. (3) Direct surface accretion onto the NS with spin-up, where $V_{\textrm{diff}}$ in Eq. 7 is the velocity difference at the NS surface.

We adopt the accretion model from Chattopadhyay et al. (Reference Chattopadhyay, Stevenson and Hurley2020) which is fitted to Figure 4 of MacLeod & Ramirez-Ruiz (Reference MacLeod and Ramirez-Ruiz2015), with $\Delta M_{\rm NS} \le 0.1 M_{\odot}$ , e.g.

(10) \begin{equation} \Delta M_{\rm NS}/M_{\odot} = a(R_{\rm comp}/R_{\odot}) + b.\end{equation}