Formation of Binary Millisecond Pulsars with Helium White Dwarfs in a New Magnetic Braking Prescription

TL;DR

This paper tackles the fine-tuning problem in forming binary MSPs with He WDs under the standard MB prescription by comparing it to the convection and rotation boosted (CARB) MB law using detailed binary evolution with the MESA code. The CARB MB framework broadens the initial-progenitor parameter space, allowing NS–MS binaries with a wide range of initial orbital periods and donor masses to evolve into detached MSP–WD systems with short orbital periods (2–9 hours) as well as MSP–WDs with longer orbits, and it can even produce persistent UCXBs. A key finding is the robust relation between the MSP–WD orbital period and WD mass, with short-period systems tending to have $M_{WD}\approx 0.16-0.17\,M_\odot$, while longer-period systems populate a broader mass range; hydrodynamic and evolutionary considerations under CARB MB also account for observed UCXBs. The work further predicts detectable low-frequency gravitational waves from detached MSP–WDs for space-based detectors (LISA, TianQin, Taiji) and provides initial-parameter-space distributions useful for population-synthesis studies. Overall, the CARB MB prescription offers a credible mechanism to form a diverse population of binary MSPs with He WDs and has testable implications for UCXB formation and GW astronomy.

Abstract

Magnetic braking (MB) mechanism plays a vital role throughout the evolution of low-mass X-ray binaries (LMXBs). Considering the standard MB prescription, the initial orbital periods of LMXBs that can evolve into binary millisecond pulsar (MSP) with He white dwarfs (WDs) and short orbital periods ($2-9~\rm hours$) are within an extremely narrow interval, which was named the fine-tuning problem. Employing the detailed binary evolution model, we investigate the evolution of LMXBs in both the standard and convection and rotation boosted (CARB) MB laws. In the standard MB case, it is difficult for donor stars to form a He core and exhaust H envelope through mass transfer at short orbital periods, making them semidetached systems. The CARB MB mechanism can drive LMXBs evolve toward compact detached MSP-WD systems in wide initial orbital periods, over which binary MSPs with long orbital periods will be produced. We obtain the initial parameter space of binary MSPs with He WDs in the initial orbital period and donor-star mass plane, which can be applied to future statistics study by population synthesis simulations. We also discuss a new relation between orbital period and WD mass, formation of persistent ultra-compact X-ray binaries with relatively long orbital periods, and detectability of compact MSP-WD systems as low-frequency gravitational wave sources.

Figures (10)

• Figure 1: Evolution of NS-MS binaries with $M_{\rm NS,i}=1.4~M_{\odot}$, $M_{\rm d,i}=1.3~M_{\odot}$, and $P_{\rm orb,i}=1.6-3.2~\rm days$ in the orbital period vs. stellar age diagram (top panel) and donor-star radius vs. donor-star mass diagram (bottom panel) under the standard MB law. The solid circles, open circles, and solid stars denote the onset of the first mass transfer, the end of the first mass transfer, and the onset of the UCXB stage, respectively. In the top panel, the horizontal dashed line and shaded area represent an orbital period of $60~\rm minutes$ and an orbital period range of $2-9~\rm hours$, respectively. In the bottom panel, the black dashed and dotted curves describe the evolutionary tracks of Roche lobe radii of binary systems with a $1.5~M_\odot$ NS in orbits with orbital periods of 9 and 2 hours, respectively.
• Figure 2: Evolution of NS-MS binaries with $M_{\rm NS,i}=1.4~M_{\odot}$ and $M_{\rm d,i}=1.3~M_{\odot}$ in Kippenhahn diagram (left axis) and orbital period vs. stellar age (right axis) diagram. The solid and open circles represent the onset and end of the first mass transfer, respectively. The He core boundary is outermost location where H1 mass fraction is less than 0.01 and He4 mass fraction is greater than 0.1. The color bar indicates the hydrogen abundance at the corresponding mass coordinate of the donor star. The light green areas and the black dashed curves represent the convection regions and the helium core masses, respectively. The red curves correspond to the evolutionary tracks of orbital periods. Top-left, top-right, bottom-left, and bottom-right panels represent systems with initial orbital periods of $2.00~\rm days$, $2.70~\rm days$, $2.93~\rm days$, and $3.20~\rm days$, respectively.
• Figure 3: Same as Figure.\ref{['s4']}, but for the evolution of four systems in $\dot{P}/P$ vs. stellar age diagram. The black solid, black dashed, black dashed-dotted, and red solid curves represent the orbital decay effect caused by standard MB, GR, mass loss (ML), and mass transfer (MT), respectively.
• Figure 4: Same as Figures \ref{['s4']} and \ref{['s5']}, but for NS-MS binaries with initial orbital periods of $P_{\rm orb,i} = 1.3$ (left panels) and $2.0~\mathrm{days}$ (right panels) in the CARB MB case.
• Figure 5: Same as Figure \ref{['s1']}, but for $P_{\rm orb,i}=1.3-25~\rm days$ and the CARB MB law.