Joint determination of the Galactic magnetar formation rate and magnetic-field decay law from the observed population

Magnetars are young, isolated neutron stars whose luminous X-ray emission, bursts, and rare giant flares are powered by the decay of ultra-strong (about 10^14-10^15 G) magnetic fields. Beniamini, Hotokezaka, van der Horst and Kouveliotou (2019) showed that the observed spin periods, spin-down rates, persistent luminosities, and supernova-remnant associations of the Galactic magnetar population jointly constrain how many neutron stars are born as magnetars and how their magnetic fields decay, but left a set of coupled population-level issues open: the exact dipole-decay law (the index $\alpha$ in $\dot B\propto B^{1+\alpha}$ and its possible variation across the population), the contribution of bursts, outbursts, and giant flares to the long-term magnetic-energy budget, the reconciliation of the high inferred birth fraction with stellar-evolution and population-synthesis predictions, the census of faint low-luminosity magnetars (observational completeness), the newborn spin-period distribution, and the detection-selection function linking intrinsic to observed samples. The authors additionally raised, as a separate individual-system question, whether complex magnetic-field-emergence scenarios operate in some magnetar-like pulsars; that question is adjacent to, but not part of, the population-level problem posed here. Later work has materially narrowed but not settled the problem: simulation-based inference (fitting a generative population model by comparing simulated populations with the observed one) on the roughly 30 known magnetars now gives a Galactic birth rate of $1.8^{+2.6}_{-0.6}$ kyr$^{-1}$, a neutron-star birth fraction of about 10.7%, and a decay slope $\alpha\approx1.9^{+0.9}_{-1.3}$; the confirmed magnetar-giant-flare sample has grown from three events (two Galactic, one in the Large Magellanic Cloud) to thirteen including extragalactic ones; and formation-channel population synthesis has been performed. The decay-law parameters, the birth fraction, the completeness-corrected census, the giant-flare share of the energy budget, and the initial-spin distribution nevertheless all remain uncertain at the factor-of-few level or worse, so the joint population-determination problem is open.

Background

Magnetars are a class of isolated neutron stars whose emission is powered not primarily by rotation or accretion but by the decay of magnetic fields far stronger than the electron critical field — the strength $B_Q\approx4.4\times10^{13}$ G at which an electron's cyclotron energy equals its rest-mass energy — with surface dipole components of order $10^{14}$–$10^{15}$ G inferred from spin-down. For an isolated neutron star spinning with period $P$ and period derivative $\dot P$, magnetic dipole braking gives a surface dipole field $B\propto(P\dot P)^{1/2}$, and magnetars cluster in the upper-right of the $P$–$\dot P$ diagram (the standard pulsar census plot of spin period against its derivative) at $P\sim2$–$12$ s. Their persistent X-ray luminosities frequently exceed their rotational spin-down power, and they produce short X/gamma-ray bursts, months-long outbursts, and extremely rare giant flares (GFs) — only three GFs had been confirmed before 2020, from the soft gamma repeaters (SGRs, the historical name for bursting magnetars) SGR 0526−66 in the Large Magellanic Cloud in 1979 and, in the Galaxy, SGR 1900+014 in 1998 and SGR 1806−20 in 2004 — each releasing up to about $10^{46}$ erg, a sizable fraction of a neutron star's magnetic free energy (the energy stored in the field above its lowest-energy configuration).

The energy source is magnetic-field decay. A standard one-parameter family models the surface dipole as $B(t)=B_0(1+\alpha t/\tau_B)^{-1/\alpha}$, equivalent to $\dot B\propto B^{1+\alpha}$ with decay timescale $\tau_B$: $\alpha=0$ is exponential decay, $\alpha=-1$ linear decay, $\alpha<0$ super-exponential decay, and $\alpha>0$ sub-exponential (power-law) decay. Physically, such decay is attributed to internal transport processes: Hall drift, in which the electron fluid flowing through the neutron-star crust advects the magnetic field, and ambipolar diffusion, the slow joint drift of magnetic field and charged particles through the stellar interior; their relative efficiencies set the observable decay index $\alpha$. The observable consequences of $\alpha$ and $\tau_B$ — the maximum spin period reached before the object fades from the magnetar population, the persistent luminosity set by the magnetic-energy loss rate, and the total number of detectable magnetars — are strongly coupled to one another and to the population's birth parameters: the Galactic formation rate, the birth-field distribution, and the birth spin $\Omega_0$.

Beniamini, Hotokezaka, van der Horst and Kouveliotou (2019; MNRAS 487, 1426) combined spin-down measurements, magnetic-energy-loss estimates probed by X-ray luminosities, and supernova-remnant (SNR) associations — stellar explosion remnants whose dynamical ages date the neutron stars inside them — to place a 2σ bound of $2.3$–$20$ magnetar births per kyr in the Galaxy, i.e. a fraction $0.4^{+0.6}_{-0.28}$ of all neutron stars born as magnetars. Fitting the observed $P$–$\dot P$ distribution, they constrained birth fields to $B_0=3\times10^{14}$–$10^{15}$ G and the decay timescale to $\tau_B\sim10^4$ yr, predicted a Galactic GF rate of about $5\times10^{-3}$ yr$^{-1}$ above $10^{46}$ erg (implying several detectable extragalactic GFs per year hidden among short gamma-ray bursts), and showed that the decay index $\alpha$ is degenerate with the number of missing faint, low-luminosity magnetars: if the observed sample is nearly complete, $\alpha\approx-1$ (linear decay), whereas if many low-luminosity magnetars are undetected, $\alpha\sim1$ would imply an intrinsic population outnumbering the observed one by up to an order of magnitude. This matters beyond neutron-star taxonomy because rapidly spinning newborn magnetars are invoked as central engines for superluminous supernovae (SLSNe), long gamma-ray bursts (GRBs), and — since the 2020 radio burst from the Galactic magnetar SGR 1935+2154 — at least some fast radio bursts (FRBs), all of which depend on the magnetar birth rate, birth fields, and birth spins.

The 2019 paper explicitly identified what remained open: (1) the functional form of the dipole decay ($\alpha$, deviations from the simple law, and possible variation across the population), tied to quantifying the faint, possibly numerous low-luminosity magnetars and hence the degree of observational incompleteness; (2) the contribution of bursts, outbursts, and rare GFs to the long-term magnetic-energy budget and to $\tau_B$, poorly constrained by only three observed GFs; (3) the origin and diversity of progenitors — the high inferred birth fraction is hard to reconcile with stellar-evolution and population-synthesis models requiring extreme progenitor masses, rotation, or pre-collapse magnetization; (4) whether magnetic-field emergence (delayed appearance of a strong internal field at the surface) explains magnetar-like pulsars such as PSR J1119−6127 and PSR J1846−0258; (5) the initial spin distribution $\Omega_0$, unconstrained by the Galactic magnetar data — in particular the fraction born spinning near break-up, needed for SLSNe and GRBs; and (6) the detection-selection function (burst-triggered versus persistent-flux detection, absorption, mission-dependent efficiencies) mapping the intrinsic population onto the observed sample, including contamination of extragalactic transient catalogs by GFs. These population-level issues — the listed items other than the out-of-scope field-emergence question (4) — are not independent: they are coupled parameters of one generative population model, and the 2019 analysis itself demonstrates the central degeneracy between $\alpha$, the missing faint population, the birth rate, and the flare budget.

Problem Statement

Determine, for the Galactic magnetar population, the birth and magnetic-evolution parameters of the underlying neutron-star population, as one joint inference: (i) the formation rate of magnetars in the Galaxy and the fraction of all neutron stars born as magnetars; (ii) the functional form of the surface-dipole decay law — the index $\alpha$ in $\dot B\propto B^{1+\alpha}$ (equivalently $B(t)=B_0(1+\alpha t/\tau_B)^{-1/\alpha}$), the timescale $\tau_B$, and whether the law varies across the population or deviates from this single-parameter family; (iii) the completeness of the current census — the number, luminosity function, and flux distribution of the faint, low-luminosity magnetars missed by existing surveys; (iv) the fraction of the long-term magnetic-energy budget released impulsively in bursts, outbursts, and giant flares, including the intrinsic per-source giant-flare rate and the contamination of extragalactic short-GRB catalogs by magnetar giant flares; (v) the newborn spin-period distribution, in particular the fraction of magnetars born spinning near break-up (the minimum spin period, of order a millisecond, at which centrifugal forces would disrupt the star); and (vi) the detection-selection function (burst-triggered versus persistent detection, absorption, mission-dependent efficiencies) linking the intrinsic population to the observed sample. (The source also raised, as a separate individual-system question, whether complex field-emergence scenarios operate in a subset of magnetar-like pulsars; that question is adjacent to, but outside, this joint population determination, which is scoped to the population-level parameters and selection function above.) The answer must show that these jointly reproduce the observed $P$–$\dot P$ distribution, persistent-luminosity function, log $N$–log $S$ counts (the cumulative number of detected sources as a function of flux), supernova-remnant age associations, and burst/giant-flare statistics, and must be reconciled with stellar-evolution, binary-population-synthesis, and magneto-thermal-transport predictions for magnetar progenitors and internal field decay. Each quantity is to be determined with quantified uncertainties tight enough to discriminate between competing decay prescriptions and between competing progenitor-channel predictions, with the independent observational constraints mutually consistent. The competing decay prescriptions are the branches of the decay law defined above — linear ($\alpha\approx-1$, consistent with a nearly complete census), exponential ($\alpha\approx0$), and sub-exponential ($\alpha\gtrsim1$, requiring a large undetected faint population); the competing progenitor channels are the single-star and binary/merger channels and the proto-neutron-star fallback-dynamo channel treated in the cited population-synthesis works. The newborn spin-period distribution (v) is, at best, bounded by the Galactic data alone, which constrain it only weakly. The values quoted above for the birth rate, birth fraction, and decay slope are the current state of the art; none of these quantities is currently known to better than a factor of a few.

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