4.1. Torque balance and (near-)equilibrium spin

We adopt $N\gt0$ for spin-up and write the accretion torque as

(2) \begin{equation}N \;=\; 2\pi I\,\dot{\nu} \;=\; -\,\frac{2\pi I}{P^{2}}\,\dot P,\end{equation}

with $I=10^{45}$ g cm $^{2}$ . For a representative $P=5.76$ h ( $2.07\times10^{4}$ s), the pre-break spin-down rate $\dot P_1=+0.68\pm0.08$ s d $^{-1}$ corresponds to $N_1\approx(\!-\!1.2\pm0.1)\times10^{32}$ dyn cm, while the post-break spin-up $\dot P_2=-2.16\pm0.35$ s d $^{-1}$ corresponds to $N_2\approx$ $(\!+\!3.6\pm0.6)\times10^{32}$ dyn cm. The corresponding spin-down power

(3) \begin{equation}L_\mathrm{sd} = 4\pi^{2} I \frac{|\kern1.5pt\dot{\!P}|}{P^{3}}\end{equation}

is ${\sim}10^{29}$ erg s $^{-1}$ for the measured P and $\dot P$ , several orders of magnitude below the X-ray luminosity. The spin evolution is therefore governed by angular-momentum exchange with the accretion flow, not by a spin-powered outflow.

The measured fractional rates, $(\dot P/P)\simeq +1.2\%$ yr $^{-1}$ before the break and $-3.8\%$ yr $^{-1}$ afterwards, imply that on year-long timescales the star can move noticeably away from and back towards a torque-balanced state. Over a decade, a steady spin-up at the observed post-break rate would reduce the period by ${\sim}38\%$ , inconsistent with the long-term behaviour seen in X-rays (Corbet et al. Reference Corbet, Sokoloski, Mukai, Markwardt and Tueller2008; Enoto et al. Reference Enoto2014). The torque reversal identified here therefore fits into a scenario in which 4U 1954+319 undergoes a sequence of spin-up and spin-down episodes of comparable magnitude, such that the long-term average torque remains small and the star resides near a quasi-equilibrium spin period determined by the wind-fed accretion flow.

4.2. Settling accretion and the magnetic field

Within the quasi-spherical settling accretion framework (Shakura et al. Reference Shakura, Postnov, Kochetkova and Hjalmarsdotter2012, Reference Shakura, Postnov, Kochetkova and Hjalmarsdotter2014; Shakura & Postnov Reference Shakura and Postnov2017), the Alfvén radius scales as

(4) \begin{equation}R_\mathrm{A}\simeq 1.5\times10^{9}\,\mu_{30}^{6/11}\,\,\dot{\!M}_{16}^{-2/11}\ \mathrm{cm},\end{equation}

where $\mu_{30}$ is the dipole moment in units of $10^{30}$ G cm $^{3}$ and $\dot M_{16}$ is the mass accretion rate in units of $10^{16}$ g s $^{-1}$ . The characteristic torque can be written as

(5) \begin{equation}N\,\sim\,\dot M\sqrt{GM_\ast R_\mathrm{A}} \propto\, \dot{\!M}^{10/11}\,\mu^{3/11},\end{equation}

so that the torque depends only weakly on both $\dot M$ and $\mu$ . On that basis, the observed torque ratio $\left|N_2/N_1\right|\approx 3.2$ can be reproduced with

(6) \begin{equation}\frac{\dot M_2}{\dot M_1}\simeq \left|\frac{N_2}{N_1}\right|^{11/10}\approx 3.6.\end{equation}

Accordingly, the observed transition can be explained by a moderate change in the mass accretion rate, without invoking a major restructuring of the flow. Variations in the sign and magnitude of the specific angular momentum carried by the wind may still contribute to the detailed torque balance. Adopting a representative low-state luminosity inferred from the low states analysed by Enoto et al. (Reference Enoto2014),

(7) \begin{equation}L_X \simeq 3\times10^{34}\left(\frac{d}{1.7\,\mathrm{kpc}}\right)^2\ \mathrm{erg\ s^{-1}},\end{equation}

the Gaia-based distance of Hinkle et al. (Reference Hinkle2020), $d \simeq 3.3\,\mathrm{kpc}$ , implies $L_X \simeq 1.1\times10^{35}\ \mathrm{erg\ s^{-1}}$ and hence

(8) \begin{equation}\dot M \simeq \frac{L_X R_\ast}{G M_\ast}\simeq 1.6\times10^{14}\left(\frac{d}{1.7\,\mathrm{kpc}}\right)^2\ \mathrm{g\ s^{-1}}\simeq 6.0\times10^{14}\ \mathrm{g\ s^{-1}},\end{equation}

for $M_\ast=1.4\,{\rm M}_\odot$ and $R_\ast=10^{6}$ cm. Although the torque analysis implies a factor-of-few increase in mass accretion rate after the break, the sector-averaged TESS flux shows no associated brightening. This is consistent with the optical continuum being dominated by the M-supergiant photosphere, for which $L_\mathrm{opt} \sim 10^{38}$ erg s $^{-1}$ (Hinkle et al. Reference Hinkle2020), compared with an accretion luminosity of order $L_\mathrm{acc}\sim10^{35}$ erg s $^{-1}$ in representative low states on the Gaia distance scale. A factor-of-few change in the accretion flow would therefore not necessarily produce a detectable change in the spin-averaged optical light curve.

For a dipole moment $\mu_{30}\sim1$ ( $B\sim10^{12}$ G) this gives $R_\mathrm{A}\approx 3\times10^{9}$ cm and a characteristic torque $N\approx 1\times10^{32}$ dyn cm, increasing to ${\sim}3.6\times10^{32}$ dyn cm when $\dot M$ is raised by a factor of ${\sim}3.6$ . Both values match, to within factors of order unity, the torques inferred from the observed $\dot P$ . Allowing for uncertainties in $L_X$ , geometry, and the settling-accretion scalings, the torque amplitude and its dependence on $\dot M$ are consistent with a surface dipole field in the range $B\approx (0.5$ – $5)\times10^{12}$ G.

For $P=5.76$ h the co-rotation radius is

(9) \begin{equation}R_\mathrm{co}=\left(\frac{GM_\ast P^{2}}{4\pi^{2}}\right)^{1/3}\approx 1.2\times10^{11}\ \mathrm{cm},\end{equation}

so that $R_\mathrm{A}\ll R_\mathrm{co}$ across the TESS baseline. The magnetosphere therefore remains well inside co-rotation, far from the centrifugal-inhibition condition $R_\mathrm{A}\approx R_\mathrm{co}$ required for a transition to a strong propeller. Magnetic or centrifugal gating models in which $R_\mathrm{A}\sim R_\mathrm{co}$ predict abrupt, correlated changes in X-ray flux and spectral hardness as the torque switches sign (Bozzo et al. Reference Bozzo, Falanga and Stella2008). We do not find clear evidence for such behaviour in the contemporaneous Swift/BAT monitoring, which shows only modest variability around the torque reversal. By contrast, brighter X-ray episodes reported at other epochs, such as the 2012 outburst (Enoto et al. Reference Enoto2014), likely probe a different accretion regime from the one sampled during the TESS baseline and may be associated with stronger spin-up phases.

The same scaling can be inverted to estimate the field strength needed to bring the magnetosphere close to $R_\mathrm{A}\approx R_\mathrm{co}$ at the observed $\dot M$ . Keeping $\dot M$ fixed and using $R_\mathrm{A}\propto \mu^{6/11}$ , raising $R_\mathrm{A}$ from $3\times10^{9}$ to $1.2\times10^{11}$ cm would require an increase in dipole moment by a factor $\sim(40)^{11/6}\approx 9\times10^{2}$ , corresponding to $B\sim 10^{15}$ G. This is well above the field range needed to match the observed torque scale and is not favoured by the absence of clear propeller episodes. A long-lived Keplerian disc is also not required by the data: for the inferred $\dot M$ and B, standard disc-torque prescriptions (see e.g. Ghosh & Lamb Reference Ghosh and Lamb1979a,b) would tend to give spin-up torques larger than observed unless the system were very close to torque equilibrium.

Our interpretation is complementary to the source-specific reanalysis of 4U 1954+319 by Bozzo et al. (Reference Bozzo, Ferrigno, Oskinova and Ducci2022a), who revisited the system in light of the revised donor properties and the Gaia-based distance. Their results highlight how the updated system parameters affect both the inferred accretion geometry and the magnetic-field constraints derived from the spin evolution. In this context, the torque reversal measured here provides an additional empirical constraint on the sign and amplitude of the torque for future source-specific modelling.

4.3. Period variability and torque noise

A purely kinematic (Doppler) origin for the observed reversal in P is strongly disfavoured by the implied radial-velocity amplitude. Interpreting the period variation ( $5.51 \to 5.72 \to 5.55$ h) as a line-of-sight velocity modulation would require a fractional amplitude $\Delta P/P\sim0.04$ . The corresponding radial-velocity amplitude,

(10) \begin{equation}v_r \simeq c\,\frac{\Delta P}{P},\end{equation}

would be of order $1.2\times10^{4}$ km s $^{-1}$ , far above the velocities expected for a wind-fed system of this type. The measured period evolution is therefore most naturally interpreted as reflecting intrinsic changes in the neutron-star spin.

Accreting pulsars generically exhibit red torque noise, with power spectra often consistent with random walks in spin frequency $\nu$ or its derivative $\dot{\nu}$ on long timescales (Deeter & Boynton Reference Deeter and Boynton1982; Bildsten et al. Reference Bildsten1997). In the settling-accretion scenario, stochastic fluctuations in the specific angular momentum of the captured wind, combined with the dynamics of the convective shell above the magnetosphere, both modulate the torque (Shakura & Postnov Reference Shakura and Postnov2017). The sector-to-sector changes in optical pulsed fraction ( ${\sim}0.1$ –1%) and the sign-changing, intermediate-strength torques observed here are consistent with stochastic angular-momentum exchange in a wind-fed accretion flow. In this interpretation, fluctuations in the captured wind and in the quasi-spherical shell can modulate the torque without requiring a persistent disc or a propeller transition (Shakura & Postnov Reference Shakura and Postnov2017). The characteristic correlation time of the noise in the TESS light curves, $\tau\sim0.2$ d, is shorter than but comparable to the spin period and to the dynamical time at radii ${\sim}10^{11}$ cm, close to the co-rotation radius. This does not establish a one-to-one causal link. It only shows that the optical red noise and the torque fluctuations occur on timescales compatible with large-scale variability in the wind-fed flow.

4.4. Comparison with other X-ray binaries

Although 4U 1954+319 is now classified as a HMXB with an M-type red supergiant donor (Hinkle et al. Reference Hinkle2020; Fortin et al. Reference Fortin2023; Neumann et al. Reference Neumann, Avakyan, Doroshenko and Santangelo2023), its spin behaviour is similar in some respects to that of SyXBs in the settling-accretion regime. GX 1+4 is the prototypical example, with extended intervals of alternating spin-up and spin-down and only modest, correlated changes in X-ray flux (e.g. González-Galán et al. Reference González-Galán2012; Ilkiewicz et al. Reference Iłkiewicz, Mikołajewska and Monard2017; Serim et al. Reference Serim, Şahiner, Çerri-Serim, İnam and Baykal2017). Its long-term evolution is interpreted as that of a NS accreting from a slow, structured giant wind and fluctuating around a torque-balanced state. 4U 1954+319 occupies a similar low-luminosity regime but at much longer spin periods and with torques of order $N\sim10^{32}$ dyn cm, well below the values reached during its bright X-ray spin-up episodes (Enoto et al. Reference Enoto2014; Corbet et al. Reference Corbet, Sokoloski, Mukai, Markwardt and Tueller2008). The sequence of spin-down followed by stronger spin-up, with comparable amplitudes and a small net torque over several years, is consistent with the generic expectations of quasi-spherical settling accretion, where moderate variations in $\dot M$ and in the specific angular momentum of the wind can drive torque reversals around an equilibrium period (e.g. Shakura et al. Reference Shakura, Postnov, Kochetkova and Hjalmarsdotter2012; Shakura et al. Reference Shakura, Postnov and Hjalmarsdotter2013; Shakura & Postnov Reference Shakura and Postnov2017; Yungelson et al. Reference Yungelson, Kuranov and Postnov2019; Mao et al. Reference Mao2024). Other symbiotic candidates such as 4U 1700+24 and Sct X-1, although lacking a secure spin period determination, are likewise interpreted as wind-fed systems with strongly magnetised NSs at low X-ray luminosities (Masetti et al. Reference Masetti, Orlandini, Palazzi, Amati and Frontera2006; Corbet et al. Reference Corbet, Sokoloski, Mukai, Markwardt and Tueller2008; Bozzo et al. Reference Bozzo, Romano, Ferrigno and Oskinova2022b). This suggests that 4U 1954+319 may sample a broader class of low-luminosity, wind-fed systems.

A similar torque phenomenology is seen in several classical wind-fed HMXBs with OB supergiant donors. Sources such as Vela X $-$ 1, 4U 1538 $-$ 52, and GX 301 $-$ 2 show strong torque noise that is statistically consistent with fluctuations driven by an inhomogeneous stellar wind (Bildsten et al. Reference Bildsten1997; Serim et al. Reference Serim, Serim and Baykal2023). The supergiant system 4U 1907+09 is a particularly close analogue: it undergoes long intervals of spin-down punctuated by transitions to spin-up, with spin derivatives of comparable magnitude and only moderate changes in luminosity (Fritz et al. Reference Fritz2006; Inam et al. Reference Inam, Şahiner and Baykal2009). Vela X $-$ 1 also shows alternating spin-up and spin-down on timescales of tens of days, with typical hard X-ray flux variations of only a factor ${\sim}$ 1.5–2 between different torque states (e.g. Martínez-Núñez et al. Reference Martínez-Núñez2017; Serim et al. Reference Serim, Serim and Baykal2023). In these systems, torque reversals and timing noise are usually linked to stochastic changes in $\dot M$ and in the angular momentum of the captured wind. A persistent Keplerian disc or a strong propeller transition is not required. The torques measured in 4U 1954+319, their amplitudes, and their sign changes are consistent with this wind-fed scenario, but at lower $L_X$ and at much longer $P_\mathrm{s}$ , extending the parameter space over which such behaviour is observed.

An observational comparison can also be made with transitional millisecond pulsars, in which coherent optical pulsations have likewise been detected while their physical origin remains debated (Ambrosino et al. Reference Ambrosino2017; Papitto et al. Reference Papitto2019; Illiano et al. Reference Illiano2023). In those systems, the optical pulsations are unexpectedly strong relative to the X-ray pulsations and have been discussed in terms of magnetospheric emission, synchrotron radiation, or accretion-related processes close to the NS. Although 4U 1954+319 differs substantially in spin period, donor type, accretion geometry, and magnetic-field regime, both cases indicate that optical timing can reveal coherent pulsations and long-term spin behaviour even when the high-energy phenomenology alone does not uniquely determine the emission mechanism.

Here the timing solution comes from the optical data. The X-ray monitoring provides context, but it does not define the spin evolution. This shows that reprocessed optical pulsations can be used to follow the spin of a wind-fed accreting pulsar in a relatively faint X-ray state. The same approach may work for other long-period wind-fed systems, especially those with spin periods of a few hours to a day and optical pulsed fractions of ${\sim}0.1$ –a few per cent. SyXBs and long-period HMXBs are good candidates because their slow spins match the cadence and baseline of all-sky optical surveys such as TESS and, in the future, PLATO (Rauer et al. Reference Rauer2014). Optical timing can complement sparse X-ray coverage and provide an additional way to monitor torque noise in wind-fed accreting pulsars. Optical timing can therefore complement sparse X-ray coverage by providing an additional way to monitor torques and torque noise in wind-fed accreting pulsars.