Comparing the Observational Properties of Rotation-Powered Binary Millisecond Pulsars with Various Companion Types

Abstract

We compare the observational properties of rotation-powered binary millisecond pulsars (BMSPs) in the Galactic Field with various companion types. First, BMSPs with diverse companion types exhibit different properties in the relation of binary orbital period versus companion mass, and in the spin period distribution of neutron stars (NSs), etc., implying multiple origins of BMSPs. Second, BMSPs with companions of CO/ONeMg white dwarfs (CO-BMSPs) show fewer sources than those with companions of Helium white dwarfs (He-BMSPs), which may result from the different evolutionary histories or accretion efficiencies in their progenitors. Third, BMSPs with main-sequence companions (MS-BMSPs) and ultra-light companions or planets (UL-BMSPs) are mostly eclipsing sources that are detected in both radio and $\gamma$-ray bands (i.e., radio+$\gamma$ sources), implying that they may be younger systems and share a faster average spin period and higher average accretion rate than CO-BMSPs/He-BMSPs. We propose that the predecessors of MS-BMSPs may share a short binary orbital distance with low-mass companion stars of $M_{\mathrm{c}}\sim 0.5$–$0.8{\mathrm{M}}_{\odot }$, which induces an efficient binary accretion process, and ultimately leaves a BMSP with a main-sequence companion due to the low efficiency of its hydrogen burning. Lastly, radio+$\gamma$ He-BMSPs share a faster average spin period of NSs than radio-only He-BMSPs. Meanwhile, these two groups of sources share similar companion mass distributions, implying the $\gamma$-ray evaporation effect may not obviously strip the companion mass of He-BMSPs during ∼0.3 Gyr, which may be due to the strong gravitational potential energy of the white dwarf companions.

1. Introduction

Millisecond pulsars (MSPs) were discovered more than forty years ago, in 1982 [1], and their fast spins can be interpreted by the recycling model [2,3]. In this scenario, a neutron star (NS) in a binary system can accrete its companion matter at a low-mass X-ray binary (LMXB) stage [4]. Then, after accreting ∼$0.1$–$0.2{\mathrm{M}}_{\odot }$ matter in ∼$0.1$–10 Gyr [5,6], the NS can be accelerated to a spin period of millisecond magnitude, and it simultaneously exhibits magnetic field strengths of $B\sim {10}^7$–${10}^9$ G [7,8,9]. The final product of the accretion-induced evolution of NS-LMXBs is a rotation-powered MSP that radiates in radio or $\gamma$-ray band [10,11,12].

Further observations have captured several pieces of evidence for the recycling scenario of MSPs: First, accretion-powered MSPs discovered through X-ray observations, e.g., SAX J1808.4-3658, detected by $RXTE$ satellite (NS spin frequency of 401 Hz), have presented a link to rotation-powered radio MSPs [13,14,15,16,17]. Second, the measurements of spin frequency derivations of NSs in LMXBs have revealed their spin-up state [16,17,18]. Third, the discovery of transitional MSPs has built a direct link between accretion-powered and rotation-powered MSPs (see, e.g., [19,20,21]). Moreover, irregular radio eclipse phenomena have been observed in some binary MSPs (BMSPs) (eclipsing BMSPs, refs. [22,23,24]), which are suggested to be related to the radiation absorption process by the ejection matter from their companions [25,26,27].

The recycling scenario for BMSPs is generally considered to occur during the post-main-sequence stage of companion stars when disk accretion takes place [6,28]. In fact, the classical binary model addresses the evolutionary process of BMSPs with a white dwarf companion [29,30,31]. Moreover, the recycling theory also considers the formation channels of isolated MSPs, which ascribes the disappearance of companions to the X-ray irradiation process in the accretion phase [32,33], or to the binary evaporation effect [34,35]. In particular, the evaporation model argues that the MSP in a binary system can ablate its companion by emitting high-energy particles/$\gamma$-rays, and ultimately leaves an isolated MSP [36]. Eclipsing BMSPs are considered to be experiencing the influence of the evaporation process [37,38]. These sources are classified as redback pulsars with non-degenerate companions, and black widow pulsars with semi-degenerate companions, and both groups of sources exhibit binary orbital periods to be less than 1 day [22,23].

Rotation-powered BMSPs have been observed with various companion types, implying the potential multiple progenitors of the systems [39,40,41]. Moreover, it is widely suggested that MSPs in the globular clusters share a quite different formation environment, origin, and evolutionary histories from those in the Galactic Field (BMSPs in the Galactic Field of our galaxy are the sources that are not located in the globular clusters) [28,42,43,44]. The ATNF pulsar catalog has recorded over 500 rotation-powered MSPs (spin period, $<10$ ms) [45], in which the fastest one shares the spin period of $P\sim 1.39$ ms [46]. About $\sim 60$% of MSPs in this catalog lie in binaries, with companions showing various types, i.e., CO, ONeMg, or Helium white dwarfs, main-sequence stars, ultra-light companions, or planets. In addition to the radio observations, $Fermi$ Gamma-ray Space Telescope has discovered about 200 rotation-powered $\gamma$-ray MSPs in the energy band around ∼$20\mathrm{MeV}$–$300\mathrm{GeV}$ [47]. It is generally found that MSPs detected with $\gamma$-rays tend to have the faster spin period on average and larger spin-down power on average than those detected without $\gamma$-rays [48,49]; therefore, $\gamma$-ray MSPs are usually considered a category of young sources. In addition, Wang, Zhang & Wang [50,51] investigated the spin period distributions of MSPs based on radio/$\gamma$-ray detections, and labeled the name radio+$\gamma$ MSPs for those detected with both radio and $\gamma$-rays, and radio-only MSPs for those detected with radio but without $\gamma$-rays. The authors found that radio+$\gamma$ and radio-only MSPs show the comparable count numbers, and the two groups of sources hold different distributions in the relation of binary orbital period versus companion mass.

It is generally proposed that binary properties, such as orbital period, companion type, and companion mass, may exert a strong influence on the formation of BMSPs [4,29]. Then, as an extension of the BMSP studies from Wang, Zhang & Wang [50,51], we try to analyze the physical properties of rotation-powered BMSPs with various companion types in the Galactic Field, and further explore their multiple origins by discussing the properties of radio/$\gamma$-ray detections. The contents of this paper are organized as follows. In Section 2, we present the selection of the BMSP samples; then, in Section 3, we analyze the physical properties of BMSPs with different companion types. In Section 4, we discuss the potential multiple origins of BMSPs and present the conclusions.

2. Samples of Binary Millisecond Pulsars

We collected rotation-powered BMSPs with spin period $P<10$ ms from the ATNF pulsar catalog1 [45], GalacticMSPs.txt2, and “Public List of LAT-Detected Gamma-Ray Pulsars”3, and further limited the samples to those with the recorded information of companion types. In addition, we only consider sources in the Galactic Field, and those in the globular clusters are omitted from the samples, since they may have experienced a more complex evolutionary process or share a diverse origin compared with those in the Galactic Field. Moreover, we collected the information of radio/$\gamma$-ray detections of these sources, as well as their companion types from the above catalogs. Furthermore, we also collected the detection properties of the irregular radio eclipses of these sources.

Then, according to the properties of companion types, radio/$\gamma$-ray detections, and radio eclipse observations, we classify the BMSP samples into different categories based on various of classification methods, i.e.,

(I)

Companion types;

(II)

Radio/$\gamma$-ray detection;

(III)

Radio eclipse observation.

And we classify these as follows:

• Class (I): According to the different companion types, the sources are classified as CO-BMSPs (BMSP with the companion of CO or ONeMg white dwarf), He-BMSPs (BMSP with the companion of Helium white dwarf), MS-BMSPs (BMSP with the companion of main-sequence star), and UL-BMSPs (BMSP with the ultra-light companion or planet, i.e., the companion mass $<0.08{\mathrm{M}}_{\odot }$).
• Class (II): According to detections in radio or $\gamma$-ray band, the sources are classified as radio+$\gamma$ BMSPs (BMSP detected with both radio and $\gamma$-ray), radio-only BMSPs (BMSP detected with radio but without $\gamma$-ray), and $\gamma$-only BMSP (BMSP detected with $\gamma$-ray but without radio, e.g., PSR J1653-0158; see [52]).
• Class (III): According to whether they are observed with an irregular radio eclipse, the sources are classified as eclipsing (redback pulsars and black widow pulsars) and non-eclipsing BMSPs.

It is noted that the above three classification methods may be mixed with each other. For example, some He-BMSPs are radio+$\gamma$ BMSPs, while the other He-BMSPs are radio-only sources.

Table 1. Rotation-powered BMSPs analyzed in this paper ^a.

Class	Subclass	Count	Sub-Count	Description
(I) b	CO-BMSP	9		BMSPs with companions of CO or ONeMg white dwarfs.
	He-BMSP	116		BMSPs with companions of Helium white dwarfs.
	MS-BMSP	16		BMSPs with companions of main-sequence stars.
	UL-BMSP	40		BMSPs with ultra-light companions or planets
				(companion mass $<0.08{\mathrm{M}}_{\odot }$).
(II) c	Radio+$\gamma$ BMSP	97		BMSPs detected with both radio and $\gamma$-ray.
	Radio-only BMSP	83		BMSPs detected with radio but without $\gamma$-ray.
	$\gamma$-only BMSP	1		BMSPs detected with $\gamma$-ray but without radio.
(III) d	Eclipsing BMSP	48		BMSPs detected with irregular radio eclipse.
	Redback pulsar		15	Eclipsing BMSPs with non-degenerate companions.
	Black widow pulsar		33	Eclipsing BMSPs with semi-degenerate companions.
	Non-eclipsing BMSP	133		BMSPs were not detected with radio eclipse.

^a BMSPs are restricted to those with spin period P < 10 ms in the Galactic Field and have the indicated companion types. ^b The classification is based on the different companion types of BMSPs. ^c The classification is based on the properties of radio/γ-ray detection of BMSPs. ^d The classification is based on the observation of radio eclipse of BMSPs. Note: See Section 2 for the details of the classifications.

lists the statistics of the collected BMSP samples. It is noted that the count number of He-BMSPs (116) is obviously larger than those of BMSPs with other companion types, and radio+$\gamma$ BMSPs (97) and radio-only BMSPs (83) share comparable count numbers.

3. Physical Property Analysis on BMSPs

It is generally suggested that binary parameters, such as orbital period, companion type, and companion mass, may be related to the formation of rotation-powered BMSPs [28,29]. Therefore, we collected the physical parameters of BMSPs in Table 1 from the ATNF pulsar catalog, including spin period (P), surface magnetic field strength (B), spin-down power ($\dot{E}$) of NS, binary orbital period ($P_{\mathrm{orb}}$), and the minimum companion mass ($M_{\mathrm{c}}$), based on which we try to analyze and discuss the physical properties of different BMSPs.

3.1. $P_{\mathrm{orb}}-M_{\mathrm{c}}$ Diagram

The relation between binary orbital period ($P_{\mathrm{orb}}$) and companion mass ($M_{\mathrm{c}}$) is widely used to trace the classification and evolution of BMSPs. Therefore, we produced a $P_{\mathrm{orb}}-M_{\mathrm{c}}$ diagram of BMSPs with the collected data and show the results in Figure 1.

First, Figure 1a shows the $P_{\mathrm{orb}}-M_{\mathrm{c}}$ relation of rotation-powered BMSPs with different companion types. It can be seen that CO-BMSPs (9) show quite a few sources, which share a $P_{\mathrm{orb}}-M_{\mathrm{c}}$ range of $P_{\mathrm{orb}}\sim 0.4$–24 day and $M_{\mathrm{c}}\sim 0.2$–$0.9{\mathrm{M}}_{\odot }$. In addition, He-BMSPs make up the major sources (115), which occupy the $P_{\mathrm{orb}}-M_{\mathrm{c}}$ range of $P_{\mathrm{orb}}\sim 0.2$–200 day and $M_{\mathrm{c}}\sim 0.06$–$0.4{\mathrm{M}}_{\odot }$. Moreover, MS-BMSPs and UL-BMSPs are located mainly in the $P_{\mathrm{orb}}$ range of $P_{\mathrm{orb}}<1$ day, and these two types of sources exhibit the different $M_{\mathrm{c}}$ ranges of $M_{\mathrm{c}}\sim 0.06$–$0.4{\mathrm{M}}_{\odot }$/$M_{\mathrm{c}}\sim 0.006$–$0.06{\mathrm{M}}_{\odot }$, respectively.

Second, Figure 1b shows the $P_{\mathrm{orb}}-M_{\mathrm{c}}$ relation of radio+$\gamma$, radio-only, and $\gamma$-only BMSPs. It can be seen that most BMSPs (26/30) with $M_{\mathrm{c}}\sim 0.006$–$0.06{\mathrm{M}}_{\odot }$ are detected as radio+$\gamma$ sources. In contrast, in the mass range of $M_{\mathrm{c}}\sim 0.06$–$0.4{\mathrm{M}}_{\odot }$, about half of BMSPs (63/134) are detected as radio+$\gamma$ sources, and the other half (71/134) are detected as radio-only sources. In addition, Figure 1c shows the $P_{\mathrm{orb}}-M_{\mathrm{c}}$ relation of eclipsing (i.e., redback pulsars and black widow pulsars) and non-eclipsing BMSPs. By comparing the distributions of Figure 1a–c, it can be noted that most MS-BMSPs are redback pulsars (12/14) and most UL-BMSPs are black widow pulsars (30/37), and these two types of sources are mostly detected as radio+$\gamma$ sources.

Third, it can be seen from Figure 1a,b that He-BMSPs present a peculiar distribution in the $P_{\mathrm{orb}}-M_{\mathrm{c}}$ relation based on the radio+$\gamma$-ray/radio-only properties. For clarity, we show the $P_{\mathrm{orb}}-M_{\mathrm{c}}$ distribution of He-BMSPs in Figure 1d, and indicate the detection properties of radio/$\gamma$-ray. It is noticed that radio+$\gamma$ He-BMSPs (51) and radio-only He-BMSPs (64) have comparable count numbers, and the two groups of sources share similar $P_{\mathrm{orb}}-M_{\mathrm{c}}$ ranges of $P_{\mathrm{orb}}\sim 0.2$–200 day and $M_{\mathrm{c}}\sim 0.06$–$0.4{\mathrm{M}}_{\odot }$.

3.2. Physical Properties of BMSPs

Rotation-powered BMSPs with various companion types exhibit different $P_{\mathrm{orb}}-M_{\mathrm{c}}$ distributions (see Figure 1a), implying the binary properties may exert different influences on the evolution of BMSPs. Then, we try to further compare other physical properties of BMSPs, e.g., NS spin periods, with different companion types.

Table 2. Physical properties of rotation-powered BMSPs with different companion types.

Category	${\langle \mathit{P}\rangle }^{\mathit{a}}$	${\tilde{\mathit{P}}}^{\mathit{b}}$	${\langle \mathit{B}\rangle }^{\mathit{c}}$	${\langle \dot{\mathit{E}}\rangle }^{\mathit{d}}$	${\langle \dot{\mathit{M}}\rangle }^{\mathit{e}}$	${\langle {\mathit{P}}_{\mathbf{orb}}\rangle }^{\mathit{f}}$	${\langle {\mathit{M}}_{\mathbf{c}}\rangle }^{\mathit{g}}$
	(ms)	(ms)	($\times {\mathbf{10}}^{\mathbf{8}}$ G)	( $\times {\mathbf{10}}^{\mathbf{34}}\mathbf{erg}·{\mathbf{s}}^{-1}$)	($\times {\mathbf{10}}^{\mathbf{16}}\mathbf{g}·{\mathbf{s}}^{-\mathbf{1}}$)	($\mathbf{day}$)	(${\mathbf{M}}_{\odot }$)
CO-BMSP	$3.2\pm 0.01$	$5.9$	$1.7\pm 0.03$	$1.1\pm 0.08$	$1.5\pm 0.08$	$8.5\pm 0.3$	0.51
He-BMSP	$3.2\pm 0.08$	$3.7$	$2.3\pm 0.09$	$0.6\pm 0.03$	$1.5\pm 0.08$	$1.3\pm 0.04$	0.19
MS-BMSP	$3.3\pm 0.3$	$2.7$	$2.6\pm 0.1$	$0.5\pm 0.09$	$1.7\pm 0.2$	$0.3\pm 0.01$	0.27
UL-BMSP	$2.3\pm 0.04$	$2.6$	$1.5\pm 0.1$	$1.6\pm 0.1$	$2.2\pm 0.2$	$0.1\pm 0.01$	0.03

^a $\langle P\rangle$: average spin period. ^b $\tilde{P}$: median value of P. ^c $\langle B\rangle$: average magnetic field strength of NS surface. ^d $\langle \dot{E}\rangle$: average spin-down power. ^e$\langle \dot{M}\rangle$: average accretion rate. ^f $\langle P_{\mathrm{orb}}\rangle$: average binary orbital period. ^g $\langle M_{\mathrm{c}}\rangle$: average minimum companion mass.

lists the average values of spin period ($\langle P\rangle$), surface magnetic field strength ($\langle B\rangle$), spin-down power ($\langle \dot{E}\rangle$) of NS, binary orbital period ($\langle P_{\mathrm{orb}}\rangle$), and minimum companion mass ($\langle M_{\mathrm{c}}\rangle$), as well as the median value ($\tilde{P}$) of P, for CO-BMSPs, He-BMSPs, MS-BMSPs, and UL-BMSPs. In addition, to trace the spin-up history of BMSPs in the accretion process, we refer to the accretion rate by the following equation [28]:

$$P=2.4\left(\mathrm{ms}\right)B_9^{6/7}{\left({\displaystyle \frac{M}{1.4{\mathrm{M}}_{\odot }}}\right)}^{-5/7}{\left({\displaystyle \frac{\dot{M}}{{\dot{M}}_{\mathrm{Edd}}}}\right)}^{-3/7}{R}_6^{18/7},$$

(1)

where $B_9={\displaystyle \frac{B}{{10}^9\mathrm{G}}}$ is the magnetic field strength in the NS surface, M is the NS mass, $\dot{M}$ is the mass accretion rate, ${\dot{M}}_{\mathrm{Edd}}={10}^{18}\mathrm{g}·{\mathrm{s}}^{-1}$ is the Eddington accretion rate, and $R_6={\displaystyle \frac{R}{{10}^6\mathrm{cm}}}$ is the NS radius. We infer the accretion rate ($\dot{M}$) of BMSPs in Table 1 with Equation (1) by assuming $M=1.4{\mathrm{M}}_{\odot }$ and $R_6=1$, and list their average values ($\langle \dot{M}\rangle$) in Table 2. For clarity, we plot the histograms of P, B, $\dot{E}$, $\dot{M}$, $P_{\mathrm{orb}}$, and $M_{\mathrm{c}}$ for CO-BMSPs, He-BMSPs, MS-BMSPs, and UL-BMSPs in Figure 2.

It is noticed from Table 2 and Figure 2 that rotation-powered BMSPs with different companion types exhibit different physical properties. First, UL-BMSPs show a smaller $\langle P\rangle$ value ($\langle P\rangle \sim 2.3\pm 0.04$ ms) than other BMSP types, and the four types of sources show decreasing median values of P from CO-BMSPs ($\tilde{P}\sim 5.9$ ms) to UL-BMSPs ($\tilde{P}\sim 2.6$ ms). Second, Kolmogorov–Smirnov ($K-S$) tests (with a 95% confidence level) indicate that neither CO-BMSPs nor He-BMSPs show similar P/$\dot{E}$ distributions to other type BMSPs; however, MS-BMSPs and UL-BMSPs share similar P/$\dot{E}$ distributions. In addition, the four types of BMSPs share the similar B distributions according to the $K-S$ tests (with a 95% confidence level). Third, CO-BMSPs/He-BMSPs share smaller $\langle \dot{M}\rangle$ values ($\langle \dot{M}\rangle \sim (1.5\pm 0.08)\times {10}^{16}\mathrm{g}·{\mathrm{s}}^{-1}$/$\langle \dot{M}\rangle \sim (1.5\pm 0.08)\times {10}^{16}\mathrm{g}·{\mathrm{s}}^{-1}$) than MS-BMSPs/UL-BMSPs ($\langle \dot{M}\rangle \sim (1.7\pm 0.2)\times {10}^{16}\mathrm{g}·{\mathrm{s}}^{-1}$/$\langle \dot{M}\rangle \sim (2.2\pm 0.2)\times {10}^{16}\mathrm{g}·{\mathrm{s}}^{-1}$). The $K-S$ tests (with a 95% confidence level) indicate that CO-BMSPs and He-BMSPs share similar $\dot{M}$ distributions, while MS-BMSPs and UL-BMSPs share similar $\dot{M}$ distributions. However, CO-BMSPs/He-BMSPs share different $\dot{M}$ distributions from MS-BMSPs/UL-BMSPs. Fourth, MS-BMSPs/UL-BMSPs show that their binary orbital periods are mostly less than 1 day, while CO-BMSPs/He-BMSPs share large $P_{\mathrm{orb}}$ ranges (both $P_{\mathrm{orb}}<1$ day and $P_{\mathrm{orb}}>1$ day). In addition, MS-BMSPs/UL-BMSPs share similar distributions of the P, B, $\dot{E}$, and $\dot{M}$ values according to the $K-S$ tests (with a 95% confidence level), but different companion types and $M_{\mathrm{c}}$ distributions. Lastly, He-BMSP/MS-BMSP share similar $\langle M_{\mathrm{c}}\rangle$ values ($\langle M_{\mathrm{c}}\rangle \sim 0.19{\mathrm{M}}_{\odot }$/$\langle M_{\mathrm{c}}\rangle \sim 0.27{\mathrm{M}}_{\odot }$), which are smaller than CO-BMSPs ($\langle M_{\mathrm{c}}\rangle \sim 0.51{\mathrm{M}}_{\odot }$) and larger than UL-BMSPs ($\langle M_{\mathrm{c}}\rangle \sim 0.03{\mathrm{M}}_{\odot }$).

3.3. Radio+$\gamma$ vs. Radio-Only He-BMSPs

It can be seen from Figure 1d that radio+$\gamma$ He-BMSPs (51) and radio-only He-BMSPs (64) show comparable count numbers, and both groups of sources share the large orbital period range of $P_{\mathrm{orb}}\sim 0.2$–200 day. Then, we try to compare the physical properties of these two types of sources, i.e., radio+$\gamma$ He-BMSPs/radio-only He-BMSPs.

Table 3. Physical properties of radio+$\gamma$ He-BMSPs/radio-only He-BMSPs.

Category	${\langle \mathit{P}\rangle }^{\mathit{a}}$	${\tilde{\mathit{P}}}^{\mathit{b}}$	${\langle \mathit{B}\rangle }^{\mathit{c}}$	${\langle \dot{\mathit{E}}\rangle }^{\mathit{d}}$	${\langle \dot{\mathit{M}}\rangle }^{\mathit{e}}$	${\langle {\mathit{P}}_{\mathbf{orb}}\rangle }^{\mathit{f}}$	${\langle {\mathit{M}}_{\mathbf{c}}\rangle }^{\mathit{g}}$
	(ms)	(ms)	($\times {\mathbf{10}}^{\mathbf{8}}$ G)	( $\times {\mathbf{10}}^{\mathbf{33}}\mathbf{erg}·{\mathbf{s}}^{-\mathbf{1}}$)	( $\times {\mathbf{10}}^{\mathbf{16}}\mathbf{g}·{\mathbf{s}}^{-\mathbf{1}}$)	( $\mathbf{day}$)	(${\mathbf{M}}_{\odot }$)
Radio+$\gamma$ He-BMSPs	$3.1\pm 0.1$	3.1	$2.1\pm 0.1$	$6.7\pm 0.6$	$1.4\pm 0.1$	$1.4\pm 0.05$	0.21
Radio-only He-BMSPs	$4.2\pm 0.1$	$4.4$	$2.7\pm 0.1$	$4.9\pm 0.3$	$1.5\pm 0.1$	$4.4\pm 0.6$	0.18

^a $\langle P\rangle$: average spin period. ^b $\tilde{P}$: median value of P. ^c $\langle B\rangle$: average magnetic field strength of NS surface. ^d $\langle \dot{E}\rangle$: average spin-down power. ^e$\langle \dot{M}\rangle$: average accretion rate. ^f $\langle P_{\mathrm{orb}}\rangle$: average binary orbital period. ^g $\langle M_{\mathrm{c}}\rangle$: average minimum companion mass.

lists the average values of spin period ($\langle P\rangle$), surface magnetic field strength ($\langle B\rangle$), spin-down power ($\langle \dot{E}\rangle$) of NS, accretion rate ($\langle \dot{M}\rangle$), binary orbital period ($\langle P_{\mathrm{orb}}\rangle$), and minimum companion mass ($\langle M_{\mathrm{c}}\rangle$), as well as the median value ($\tilde{P}$) of P, for radio+$\gamma$ He-BMSPs/radio-only He-BMSPs. In addition, Figure 3 shows the histograms of these parameters. It is noticed that radio+$\gamma$ He-BMSPs share smaller $\langle P\rangle$ values ($\langle P\rangle \sim (3.1\pm 0.1)$ ms), smaller $\tilde{P}$ values ($\tilde{P}\sim 3.1$ ms), and larger $\langle \dot{E}\rangle$ values ($\langle \dot{E}\rangle \sim (6.7\pm 0.6)\times {10}^{33}\mathrm{erg}·{\mathrm{s}}^{-1}$) than radio-only He-BMSPs ($\langle P\rangle \sim (4.2\pm 0.1)$ ms, $\tilde{P}\sim 4.4$ ms, $\langle \dot{E}\rangle \sim (4.9\pm 0.3)\times {10}^{33}\mathrm{erg}·{\mathrm{s}}^{-1}$), while the two types of sources share similar values of $\langle B\rangle$ (∼$(2.1\pm 0.1)\times {10}^8\mathrm{G}$/$(2.7\pm 0.1)\times {10}^8\mathrm{G}$), $\langle \dot{M}\rangle$ (∼$(1.4\pm 0.1)\times {10}^{16}\mathrm{g}·{\mathrm{s}}^{-1}$/$(1.5\pm 0.1)\times {10}^{16}\mathrm{g}·{\mathrm{s}}^{-1}$), and $\langle M_{\mathrm{c}}\rangle$ (∼$0.21{\mathrm{M}}_{\odot }$/$0.18{\mathrm{M}}_{\odot }$).

Furthermore, we use the $K-S$ test to compare the physical parameter distributions between radio+$\gamma$ and radio-only He-BMSPs.

Table 4. Results of $K-S$ tests for He-BMSPs.

Category	Count	(p-Value)	Reject ${{\mathit{H}}_0}^{\mathit{a}}$
Spin Period—P
Radio+$\gamma$ He-BMSPs	52	$7.77\times {10}^{-9}$	Yes
Radio-only He-BMSPs	64
Magnetic Field Strength of NS Surface—B
Radio+$\gamma$ He-BMSPs	50	$2.62\times {10}^{-1}$	No
Radio-only He-BMSPs	62
Spin-down Power—$\dot{E}$
Radio+$\gamma$ He-BMSPs	50	$6.91\times {10}^{-8}$	Yes
Radio-only He-BMSPs	62
Accretion Rate—$\dot{M}$
Radio+$\gamma$ He-BMSPs	50	$9.34\times {10}^{-3}$	Yes
Radio-only He-BMSPs	62
Binary Orbital Period—$P_{\mathrm{orb}}$
Radio+$\gamma$ He-BMSPs	52	$8.96\times {10}^{-2}$	No
Radio-only He-BMSPs	64
Minimum Companion Mass—$M_{\mathrm{c}}$
Radio+$\gamma$ He-BMSPs	51	$6.02\times {10}^{-2}$	No
Radio-only He-BMSPs	64

^a $H_0$: the null hypothesis by assuming the two groups of data share the same continuous distribution (the confidence level is set at 95%).

shows the test results, suggesting that the two groups of sources share the different distributions of the P, $\dot{E}$, and $\dot{M}$ values with a 95% confidence level, while the B, $P_{\mathrm{orb}}$, and $M_{\mathrm{c}}$ values of the two types of sources come from the same distribution.

As a precaution, we recall the assumptions and reliability of the physical parameters compared above: The P and $P_{\mathrm{orb}}$ values are directly measured parameters, based on which the comparison between the different BMSP types is relatively reliable. However, the B, $\dot{E}$, $\dot{M}$, and $M_{\mathrm{c}}$ values are derived parameters based on a series of theoretical assumptions, which are calculated through the directly measured parameters. For example, the surface magnetic field strength of NS (B) is calculated under the assumption of a magnetic dipole radiation model with an NS mass of $1.4{\mathrm{M}}_{\odot }$ and radius of 10 km [45]. Another example is the minimum companion mass ($M_{\mathrm{c}}$), which is calculated by assuming an inclination angle of $i=90$ degrees and an NS mass of $1.35{\mathrm{M}}_{\odot }$, i.e., by assuming the orbit to be perpendicular to the Earth [45], which might differ from the true value. Therefore, there may exist uncertainties and observational selection effects when comparing these derived parameters between the different BMSP types, though these parameter values are calculated under the same theoretical assumptions for the different BMSP types. However, it is noticed that the four BMSP types (CO-BMSPs, He-BMSPs, MS-BMSPs and UL-BMSPs) share different distributions of P and $P_{\mathrm{orb}}$ values, i.e., the directly measured parameters, implying that BMSPs with various companion types may share different evolutionary histories.

4. Discussion and Conclusions

In order to explore the origin of rotation-powered BMSPs with various companion types in the Galactic Field, i.e., CO-BMSPs, He-BMSPs, MS-BMSPs, and UL-BMSPs, we investigate the distributions of their physical parameters, including spin period, surface magnetic field strength, spin-down power of NS, accretion rate, binary orbital period, minimum companion mass, and further discuss the properties of radio/$\gamma$-ray detections. The conclusions obtained are listed below:

• It is widely accepted in theory that MSPs are formed through the accretion-induced spin-up process in LMXBs [2,4]. However, considering the complexity of binary evolution, it is usually suggested that MSPs may have multiple origins [39,41], which could depend on the orbital and companion properties of LMXBs. In fact, binary properties, such as orbital period, companion type and companion mass, may exert the significant influence on the formation of MSPs, which arises the diverse observational features of MSPs [4,29]. Generally, the formation as well as evolution of BMSPs can be traced by the observed correlation between binary orbital period and companion mass ($P_{\mathrm{orb}}-M_{\mathrm{c}}$ diagram; see also [53,54,55]). Then, BMSPs with various companion types, i.e., CO-BMSPs, He-BMSPs, MS-BMSPs, and UL-BMSPs, have different $P_{\mathrm{orb}}-M_{\mathrm{c}}$ distributions (see Figure 1a), which implies that they may share different evolutionary histories. In addition, these four groups of BMSPs also exhibit different distributions of the spin period (P) and spin-down power ($\dot{E}$) of NS, accretion rate ($\dot{M}$), binary orbital period ($P_{\mathrm{orb}}$), and minimum companion mass ($M_{\mathrm{c}}$) (see Table 2 and Figure 2), implying that the binary properties and companion features can affect the formation and evolution of BMSPs, which also support their multiple-origin scenario.
• In BMSP samples with spin period $P<10$ ms, CO-BMSPs (9) show quite a few sources; as a comparison, He-BMSPs (116) occupy the major part of BMSP sources (see Table 1). In theory, a solar-metallicity isolated main-sequence star with mass $M⩽8{\mathrm{M}}_{\odot }$ will evolve into a CO white dwarf with $M\sim 0.6$–$0.8{\mathrm{M}}_{\odot }$, or an ONe white dwarf with $M\sim 1.0{\mathrm{M}}_{\odot }$. However, the formation of a low-mass He white dwarf ($M<0.5{\mathrm{M}}_{\odot }$) needs the progenitor star to lose most of the mass on the red giant branch. This process can occur in close binaries, where a companion can strip the outer envelope of its low-mass white dwarf progenitor before it begins the helium burning [56]. In particular, the NS in a binary needs to accrete ∼$0.1$–$0.2{\mathrm{M}}_{\odot }$ matter from its companion in the ∼$0.1$–10 Gyr LMXB phase so as to form an MSP; then, its companion may lose most of the mass during this phase through the accretion or X-ray irradiation process, where the strong interaction in binary may cause the system to leave a He white dwarf companion, but not a CO white dwarf or ONe white dwarf companion. The above may be one of the explanations for the count number of CO-BMSPs with $P<10$ ms being lower than that of He-BMSPs. In addition, the less observed CO-BMSPs with $P<10$ ms may also result from the low accretion efficiency of their progenitors, i.e., the progenitors of some CO-BMSPs in LMXB phase may have a low accretion efficiency, causing their NSs to not accrete enough companion matter to spin-up to a period of $P<10$ ms. Moreover, the more observed He-BMSPs may also be affected by the observational selection effects. In fact, some studies indicate that most BMSPs containing white dwarfs may have experienced a long-lived stable mass transfer phase during their progenitor systems, which increases the chance of observing a BMSP with a Helium white dwarf [57,58,59].
• It is noticed from Figure 1a–c that most MS-BMSPs and UL-BMSPs are eclipsing and radio+$\gamma$ sources, implying that they are young rotation-powered BMSPs. And MS-BMSPs/UL-BMSPs share the faster median value of the NS spin period ($\tilde{P}$), and higher average accretion rate ($\langle \dot{M}\rangle$) than those of CO-BMSPs/He-BMSPs (see Table 2 and Figure 2), implying that the predecessors of MS-BMSPs/UL-BMSPs may have had the higher accretion efficiency. In addition, the short binary distance of MS-BMSPs/UL-BMSPs implies that their progenitors may have experienced a strong binary interaction, e.g., accretion, X-ray irradiation, or $\gamma$-ray evaporation process, and ultimately leaves a pulsar with a fast spin and an ultra-light companion, like an UL-BMSP. Furthermore, according to the spin-up theory, the NS needs to experience a time-scale of ∼$0.1$–10 Gyr accretion process at the LMXB stage to form an MSP [4]. Then, the question is how do the main-sequence companions of MS-BMSPs survive during the binary accretion process. Some authors have proposed the potential formation channels for these systems, such as through the dynamical processes in a globular cluster (e.g., [60]) or via a triple system (e.g., [61]). Here, we propose that the predecessors of MS-BMSPs may have the short binary orbital distance with the low-mass star of $M_{\mathrm{c}}\sim 0.5$–$0.8{\mathrm{M}}_{\odot }$ as the companion. This low-mass companion may have the low efficiency of hydrogen burning [62,63,64], and age longer than the accretion time-scale of ∼$0.1$–10 Gyr; then, the NS can capture and accrete the companion matter efficiently due to the short binary distance. Finally, the NS will evolve into an MSP by accreting ∼$0.1$–$0.2{\mathrm{M}}_{\odot }$ matter, and its companion still keeps it as a main-sequence star.
• Radio+$\gamma$ He-BMSPs (52) and radio-only He-BMSPs (64) share comparable count numbers (see Figure 1d and Figure 3), and radio+$\gamma$ He-BMSPs share a faster average spin period ($\langle P\rangle$) (∼$3.1\pm 0.1$ ms) and larger average spin-down power ($\langle \dot{E}\rangle$) (∼$(6.7\pm 0.6)\times {10}^{33}\mathrm{erg}·{\mathrm{s}}^{-1}$) than radio-only He-BMSPs ($\langle P\rangle \sim 4.2\pm 0.1$ ms and $\langle \dot{E}\rangle \sim (4.9\pm 0.3)\times {10}^{33}\mathrm{erg}·{\mathrm{s}}^{-1}$; see Table 3). Moreover, the $K-S$ tests indicate that these two groups of sources share the different distributions of P, $\dot{E}$ and $\dot{M}$ (see Table 4). However, considering the fact that radio+$\gamma$ He-BMSPs/radio-only He-BMSPs share similar binary orbital period ranges of $P_{\mathrm{orb}}\sim 0.2$–200 day, the same companion types, and similar companion mass ranges of $M_{\mathrm{c}}\sim 0.06$–$0.4{\mathrm{M}}_{\odot }$ (see Figure 1d), we propose that the difference between the two types of sources may not be dominated by the influence of binary evolution but by the intrinsic properties of MSPs. Theoretically, the $\gamma$-ray luminosity ($L_{\gamma }$) of a pulsar is predicted to be affected by its threshold voltage ($\Phi$) in the radiation region [65,66], which further links to its spin-down power ($\dot{E}$), i.e., $L_{\gamma }\propto \Phi \propto {\dot{E}}^{1/2}\propto P^{-3/2}$. The prediction of $L_{\gamma }\propto {\dot{E}}^{1/2}$ is basically consistent with $Fermi$ observations; therefore, He-BMSPs with faster spin periods and larger spin-down powers may hold larger $\gamma$-ray luminosity, which are more likely to be detected as radio+$\gamma$ sources. Furthermore, radio+$\gamma$ He-BMSPs share an average spin period ($\langle P\rangle \sim$$3.1\pm 0.1$ ms) shorter than those of radio-only He-BMSPs ($\langle P\rangle \sim 4.2\pm 0.1$ ms) by ∼1 ms (see Table 3). Then, this spin difference can infer a time-scale of $\tau \sim \mathrm{\Delta }P/\dot{P}\sim 1\times {10}^{-3}\mathrm{s}/{10}^{-19}\mathrm{s}·{\mathrm{s}}^{-1}\sim 0.3$ Gyr by considering the typical spin-down rate of $\dot{P}\sim {10}^{-19}\mathrm{s}·{\mathrm{s}}^{-1}$. This result infers a possible evolutionary process: a young rotation-powered He-BMSP with a short spin period can emit signals in both radio and $\gamma$-ray bands. Furthermore, after a spin-down evolution with a time-scale of ∼$0.3$ Gyr, its spin period lengthens by ∼1 ms, and its $\gamma$-ray luminosity decreases so that this source can only be detected with radio but without $\gamma$-ray. Lastly, it should also be noticed that radio+$\gamma$ He-BMSPs/radio-only He-BMSPs share similar distributions of the companion masses (see the $K-S$ test results in Table 4), implying that the $\gamma$-ray evaporation effect may not obviously strip the companion mass of He-BMSPs in the ∼$0.3$ Gyr evolutionary process. Theoretically, the gravitational potential energy around the white dwarf surface is too high to allow efficient evaporation, if comparing with that of a planet. In addition, the white dwarf has a very small radius, compared with a star, such that it absorbs a small fraction of the $\gamma$-ray flux from an MSP; therefore, it would be evaporated insufficiently. Here, we also recall the influence of the observational selection biases on the conclusion of radio+$\gamma$ and radio-only BMSPs. In fact, besides the blind search, the discovery of the $\gamma$-ray pulsars is usually based on the cross-certification (ensuring a precise source position) between the ephemerides of $\gamma$-ray pulsars and radio (or X-ray/optical) pulsars. Therefore, the presence of a radio counterpart plays a vital role in identifying $\gamma$-ray pulsars [48], making the detection of two samples (radio+$\gamma$ and radio-only) be not independent. Moreover, the difference between radio+$\gamma$ and radio-only He-BMSPs are also affected by the sensitivity of the telescope for the radio and $\gamma$-ray detections. In fact, $Fermi$-LAT shares the sensitivity range of $\gamma$-ray of ∼$20\mathrm{MeV}$–$300\mathrm{GeV}$, which defines radio+$\gamma$ and radio-only MSPs in this paper. In the future, as the enhance of the sensitivity of the radio and $\gamma$-ray detections, more radio and $\gamma$-ray He-MSPs may be discovered.