Sco X-1 as a continuous gravitational waves source: modelling the secular evolution using MESA

TL;DR

This work integrates long-term binary evolution with neutron-star spin and non-axisymmetric deformations to evaluate Sco X-1 as a continuous gravitational-wave source. By coupling MESA-based binary evolution with two deformation mechanisms—magnetic mountains and crustal breakage—the authors map how accretion history and donor-star properties influence GW amplitudes, frequencies, and detectability. They find that current detectors require relatively large ellipticities ($\varepsilon\gtrsim10^{-6}$) for Sco X-1 to be detectable, whereas third-generation observatories could probe $\varepsilon$ down to $\sim6\times10^{-9}$ and detect a substantial fraction of Sco X-1-like systems, with detectability strongly dependent on accretion efficiency $\eta$ and donor mass $M^d$. The results demonstrate that the nominal torque-balance expectation underestimates the potential GW signal in many scenarios and provide a framework to interpret and plan future continuous GW searches for accreting neutron stars.

Abstract

We study the prospects for detecting continuous gravitational waves (GWs) from Sco X-1 and evaluate the most likely waveform and progenitor parameters. We model the spin of the neutron star by the accretion torque and the gravitational-wave torque, considering two mechanisms for generating the non-axisymmetry responsible for the latter: magnetic mountains and crustal breakage deformation. Both torques are intertwined with the binary evolution, which we trace from the formation of the NS in a binary system with a main-sequence companion. We do this with MESA, starting from a set of initial binary configurations. At current sensitivity, a magnetic ellipticity of $\varepsilon\gtrsim 10^{-6}$ is necessary for detection. The highest frequency at which we have detectable signals increases with the accretion efficiency $η$, and it can be as high as 360Hz. At 3G (Cosmic Explorer/Einstein telescope) sensitivity, less deformed Sco X-1 NSs, with ellipticities as small as $6\cdot 10^{-9}$, are detectable, but the waveform highly depends on the binary system: the highest frequency of detectable signals spans the very broad range 600-1700Hz, strongly depending on $η$ and mass of the progenitor donor star $M^d$. If $η\leq$30%, the crust does not break. For $η\in$[40%,60%] only progenitors with $M^d\geq[1.1,1.5]M_{\odot}$ present crustal breakage, while if $η\geq$70% all crusts break. In some systems, the crust breaks during their Sco X-1 phase. If Sco X-1 were one of those systems, it would be emitting a very loud GW signal sweeping from O(1000)Hz down to torque-balance frequencies in $\approx 150000[\varepsilon /10^{-5}]^{-2/5}$ years. We estimate the current detection probability for this signal to be under 1%; this probability increases substantially - to around 41% - with 3G detectors.

Figures (8)

• Figure 1: Orbital period and mass ratio of a Sco X-1 progenitor system as it evolves -- the arrow indicates the flow of time. The color-code indicates the effective temperature of the donor, and temperatures non-compatible with Sco X-1 are solid yellow. The red rectangle defines the Sco X-1 compatible region. $t_1$ marks the time when the system first reaches a stable accretion rate of $10^{-8} \, M_{\odot} \cdot \textrm{yr}^{-1}$. $t_2$ is the earliest time when the system is Sco X-1 compatible. For this illustrative system $\eta=50\%$ accretion efficiency.
• Figure 2: The three rows of plots differ for the value of the accretion efficiency: 30%, 50% and 80%, and refer to the same progenitor system defined by: $P=1.91 \, \textrm{d}, M^{d}=1.25 \, M_{\odot}$. On the left we have the spin frequency evolution of the neutron star in Sco X-1 in the case of high deformation (solid blue curve) and small deformation (dashed blue curve), in both cases due to magnetic confinement of accreted matter. The twin $y$ axis shows the moment of inertia as a function of time (orange curve). We highlight the evolutionary stage where the system resembles Sco X-1 with scatter markers superimposed on the curves (in this example, in green). On the right, we plot the gravitational wave amplitude $h_0(f)$ emitted by the highly and weakly deformed neutron star (same as the left-side plots). We compare it with the most recent gravitational wave amplitude upper limits $h_0^{ul}$ (2022ApJ...941L..30A, solid grey curve), as well as with the nominal torque-balance condition introduced in Section \ref{['sec:torques']}. The $h_0(f)$ curves present a "turning point" in the case of high ellipticity: very early on, the system reaches torque balance, so the frequency stops increasing. However, since the mass accretion rate $\dot{M}_\textrm{a}$ decreases over time, the accretion torque also decreases, and the neutron star begins to slowly spin down. Even though the moment of inertia $I$ increases, since $h_0 \propto I \, f^2$, overall $h_0$ decreases, creating the "turning point".
• Figure 3: Same as Figure \ref{['fig:evolution_example']} but in each row of plots we vary the progenitor system, maintaining the accretion efficiency constant at $\eta=50\%$. Generally, the spin-up starts right at the onset of mass transfer, so we can compare the spin-evolution phases to see how the length of the mass transfer phase increases as the mass of the donor star increases. Notice the left plot in the third row: at $t=2.325 \times 10^9 \, \textrm{yr}$ the spin-up has not yet started but the moment of inertia is already above the initial value of $\approx 1.45 \cdot 10^{45} \textrm{g} \cdot \textrm{cm}^2$. This is because, for a certain period of time, the mass transfer rate is non-zero but the disk-sustained condition of Equation \ref{['eq:dubuis']} does not hold and the accretion torque is zero; the neutron star can however accrete mass, increasing its moment of inertia. Progenitor donor stars with masses greater than $1.5 \, M_{\odot}$ are omitted from the figure because their multiple accretion episodes introduce complexities that hinder the intended interpretability of the visualization. See 2025AA...693A.314M for an explanation of why multiple Roche lobe overflow phases happen in systems with heavy donors.
• Figure 4: The $M^d-\eta$ space color-coded with the maximum frequency of continuous gravitational waves detectable by $3G$ detectors. If we do not find Sco X-1 compatible configurations the bin is left empty (white). The pattern shown in this plot can also be appreciated by looking at the summary Figure \ref{['fig:h0_vs_f_mag']} in appendix \ref{['sec:appendix_figs']}.
• Figure 5: Spin evolution (solid lines, left plot) and concurrent gravitational wave emission (right plot) of a Sco X-1 neutron star going through crustal failure, which leaves behind a residual ellipticity of $\varepsilon = 10^{-5}$, for three values of the accretion efficiency: 30%, 40% and 50%; among these, only $\eta=50\%$ leads to crustal failure therefore to continuous gravitational wave emission. The dashed lines on the left plot represent $45\%$ of the Keplerian break-up spin frequency as a function of time. The gravitational wave "trace" of the right plot is color-coded with the instantaneous gravitational wave frequency time derivative. It takes $\approx 250 \, \textrm{yr}$ to spin-down from A, the point of crustal breakage, to B, where spin frequency is half of the break-up frequency. It takes further $\approx 3500 \, \textrm{yr}$ to reach C, where the spin frequency is reduced by another factor of 2. In D, reached from C in $\approx 200 \, 000 \, \textrm{yr}$, the neutron star is in torque balance, persisting there for slightly less than a million years before leaving the Sco X-1 compatibility state. The progenitor system considered here is the same as the one considered in Figure \ref{['fig:evolution_example']}.