Processing math: 52%
2021 Impact Factor 2.944 Impact factor of Chinese Physics C is 3.6 in 2023 Impact factor of Chinese Physics C is 3.6 in 2022 2021 Impact Factor 2.944 Impact factor of Chinese Physics C is 3.6 in 2023
Advanced Search
Chinese Physics C> 2023, Vol. 47> Issue(8) : 084106 DOI: 10.1088/1674-1137/acdc8b

New insights into the limit of the magnetic monopole flux and the heating source in white dwarfs

  •  College of Science, Hainan Tropical Ocean University, Sanya 572022, China

  • Based on the magnetic monopole (MM) catalytic nuclear decay (Rubakov-Callan (RC) effect), we propose five new models to discuss the limit of the MM flux and the heating energy resources of white dwarfs (WDs) based on observations of 13 red giant branch (RGB) stars. We find that the number of MMs captured can reach a maximum value of 9.1223×1024 when m=1017 GeV, nB=5.99×1031 cm−3, ϕ=7.59×10−26cm−2s−1sr−1. The good agreement of our calculated luminosities for WDs with observation provides support for our model based on the RC effect by MMs. We obtain a new limit of the MM flux of ξ=ϕ⟨σmvT⟩−28≤9.0935×10−13cm−2s−1sr−1, and ξ≤4.9950×10−13cm−2s−1sr−1 at nB=5.99×1031cm−3 when m=1015GeV, β=9.4868×10−3, and m=1017GeV, β=10−3, respectively. Our results show that the RC effect could cause heating that prevents white dwarfs from cooling down into a stellar graveyard. Our results will also provide a new idea for further research on the upper limit of MM flow (note: nB,σm,m,ϕ,ξ are the baryon number density, reaction cross section, mass, MM flux, and the new limit of the MM flux, respectively, and β=vT/c is the ratio of the speed of MMs to that of light).

  • I. INTRODUCTION

    White dwarfs (hereafter WDs) are usually made up of C+O. However, it is also possible for their cores to be hot enough to burn carbon but not hot enough to burn Ne, forming a WD with a core of O+Ne+Mg. At the later stage of WDs, the star ejects large quantities of matter. After great mass loss, if the remaining core mass is less than 1.44 solar masses, the star may evolve into a WD.

    WDs form at very high temperatures. Because they have no source of energy, they will therefore gradually give off heat and cool down, whereupon its radiation will decrease over time from its initial high color temperature to red. This surface temperature is defined in astronomy as the effective temperature Teff as per Stefan's law, so that

    Lrad=4πR2σT4eff,

    		(1)

    where R the radius of the star, Lrad is the radiation luminosity, and σ=5.6704×10−5ergs−1cm−2K−4 is the radiation constant from Stefan's law. Teff is a measure of the energy flux at the surface and not a real temperature, but it nevertheless constitutes a useful measure of the atmospheric temperature of the star.

    As is well known, the effective temperature of WDs is mostly in the range 5500–40000 K, while a few are outside this range, and the internal temperature of WDs is on the order of ∼106−107K, with a total thermal energy less than 1047ergs. Mestel. [1] discussed the energy sources of WDs. Avakian. [2] also studied the configurations of hot WDs with nuclear sources of energy. Bildsten & Hall. [3] discussed the sources of white dwarfs, suggesting that there are small amounts of 22Ne in some WDs that may constitute an extra source of heat in carbon-oxygen WDs. Single-particle 22Ne sedimentation may be considered a possible heat source [4, 5]. However, some work suggests that 22Ne must separate into clusters, enhancing diffusion, in order for sedimentation to provide heating on the observed timescale. Recently, the sources of ultra-high-energy photons for WD pulsars have been discussed by Lobato et al. [6]. Cheng et al. [7] discuss the cooling anomaly of high-mass WDs, pointing out that 22Ne settling in C/O-core WDs could account for this extra cooling delay. Caplan et al. [8] studied this topic using molecular dynamics methods and phase diagrams, from which they ruled out the isotope 22Ne as a possible cause of the extra heating. Therefore, the problem of additional heat sources for WDs remains a challenging topic.

    In this paper, we selected 13 red giant branch (RGB) stars to present a new model of the number of magnetic monopoles (hereafter MMs) captured, and seek to solve the energy source problem for WDs based on MM catalytic nuclear decay (the Rubakov-Callan (RC) effect) [9, 10]. MMs are hypothetical magnetic particles with a single north or south magnetic pole, which have been proposed in string theory. Research on MMs has long been a hot topic among physicists and astronomers. Some papers have discussed the issues of MMs, (e.g., Callan [9], Detrixhe et al. [11], Frank et al. [12], Fujii & Pierre [13], Kain [14], Rajantie [15]). Recently, Mavromatos & Mitsou. [16] discussed the developments in both theory and experimental searches for MMs in past, current, and future colliders and in the cosmos. We also are interested in the problem of MMs and other related issues ((e.g., Liu. [17, 18], Liu & Gu [19], Liu et al. [20], Liu [21−26], Peng et al. [27, 28]).

    The arrangement of this paper is as follows. In the next section, we discuss the number of possible magnetic monopoles in space and the luminosity due to the RC effect by MMs. In Section III, we describe our models and the luminosity function due to magnetic monopole catalytic nuclear decay. In Section IV, some results and discussions are presented. Finally, our conclusions are summarized in Section V.

    II. THE NUMBER OF MMs CAPTURED IN SPACE AND THE LUMINOSITY DUE TO THE RC EFFECT BY MMs

    According to some research, the interaction of MMs with neutral hydrogen atoms is very weak. Therefore, during the process of formation of celestial bodies, very few MMs are captured in the collapse of a neutral hydrogen cloud and collect in the core of a star or planet. MMs typically may be contained within stars and planets, and they are mostly captured from space during their lifetime after formation. One type of interaction that MMs undergo in stars and planets might be the RC effect, through which MMs may catalyze nucleon decay, as expressed by p+M→e+π0+M+debris (x%) and p+M→e+μ±+M+debris (y%). The ratio of the cross sections of the above reactions x/y may be a few percent, which is on the order of ∼10−4 [10]. Bernreuther and Craigie [29] discussed the cross section of monopole-induced proton decay in SU(5) for the above reactions, showing that the ratio of cross sections of the above reactions x/y may be (2.5∼2.8)×10−4/(0.1∼0.3).

    The number (flow) of MMs captured in space at the surface of stars and planets (including earth) is estimated as follows [30]:

    Nm=4π2R2ηϕt[1+(vescvm)2],

    		(2)

    where ϕ is the flux of MMs intercepted in space and vesc=(2GM/R)1/2, t are the escape velocity from the star and cooling age of the star, respectively. vm represents the velocity of MMs in space, For a planet, vescvm≪1, and vescvm≈1 for ordinary stars, but vescvm≫1 for compact objects such as WDs, neutron stars, galactic nuclei, and quasars. Therefore, the ratio between the number of MMs captured and the number of nuclei in the MMs accumulation area is [30]

    NmNB=5.10×10−34ηR2∗(t9M∗)(ϕϕ0)[1+4.256v−2−3(M∗R∗)],

    		(3)

    where R∗=R/R⊙, M∗=M/M⊙, v−3=vm/10−3c, t9=t/109Yr, and c is the speed of light. ϕ0≈10−12cm−2s−1Sr−1 [31]. η is the probability of the capture of MMs by stars, which depends on the ratio of the penetration distance lpd of an MM in a star to the star's radius. In general, we have lpd≈1.2×1030v−3n−1eT1/2e for plasma [30]. For example, with ne∼1022cm−3,Te∼106K for the sun, we have lpd∼1011,η∼0.7; however, for WDs and neutron stars, ne≥1030,ne≥1035cm−3, respectively, and η∼1. For quasars and active galactic nuclei, M∗∼108,Te∼105K, η∼1.

    The velocity of MMs is also determined as a function of monopole mass by βT as follows [32].

    vm=cβT={3×10−3c(1016GeVm)0.5(m<1017GeV)10−3c(otherwise).

    		(4)

    According to Eq. (3), for WDs we have η∼1; thus, the total number of magnetic monopoles trapped in space after the formation of stars (or planets) is estimated to be

    Nm=7.18×1011nBR5∗ϕ(t9M∗)[1+4.256v−2−3(M∗R∗)].

    		(5)

    For astronomy, the most important property of a magnetic monopole is that it can trigger the RC effect, as independently proposed by Rubakov and Callen [9, 10]. The reaction cross section is about σm≈10−25∼10−26cm2, almost reaching the Thomson cross section (6.665×10−25cm2). The luminosity of various types of celestial bodiesdue to the RC effect (i.e., RC luminosity) can be estimated as follows. In the core area, where the magnetic monopole is concentrated, the nuclear decay reaction is catalyzed by the magnetic monopoles and the total luminosity produced is [30]

    Lm≈4π3r3cnmnB⟨σmvT⟩mBc2=NmnB⟨σmvT⟩mBc2,

    		(6)

    where rc, and nm,nB are the radius of the stellar central region and the number densities of MMs and nucleons, respectively.

    In Eq. (6), vT is the thermal movement speed of the nucleus relative to the magnetic monopole. We will ignore the thermal velocity of the magnetic monopole due to its great mass. We thus only consider the contributions from the thermal velocity of the nucleus. According to 1/2mv2T=3/2kT, we have vT=√3kT/mB≈1.5745×107T1/26cm/s, where T is the temperature, T6=T/106K, k=1.38×10−16erg/s is the Boltzmann constant, and mB≈1.67×10−24g is the nucleon mass. As the central temperature of WDs is about ∼106K, we have vT∼10−3c.

    As a general rule, the reaction cross section of the RC effect σm is 10−26∼10−24cm2. Ma & Tang [33] gave a value σm≈4.28676×10−24cm2 for the cross section of the proton with different channels using SU(5) grand unification theory. In the RC process, MMs induced nucleon decay, followed by nucleon decay into π0 mesons, μ± leptons, and positrons e+, and μ± and π0 again decay into photons and electron-positron pairs e±. The positrons then undergo annihilation with electrons to form photons. The net effect is that the rest mass energy of nucleons (mBc2) is entirely converted to radiant energy with 100% efficiency (1mBc2≈1GeV≈1.6×10−3ergs).

    III. THE MAGNETIC MONOPOLE MODEL AND RC LUMINOSITY INSIDE WHITE DWARFS

    A The mass and radius for the red giant branch (RGB) phase

    It is well known that WDs evolve from red giants with mass less than 8M⊙. The MM content of the trapping accumulation set is mainly from the red giant branch (RGB) phase. The M∗ and R∗ for RGB stars is given by [34, 35]

    Mr∗=MRGM⊙=(νmaxνmax,⊙)3(ΔνfΔνΔν⊙)−4(TeffTeff,⊙)1.5,

    		(7)

    Rr∗=RRGR⊙=(νmaxνmax,⊙)(ΔνfΔνΔν⊙)−2(TeffTeff,⊙)0.5,

    		(8)

    where νmax,⊙=3090μHz, Δν⊙=135.1μHz, Teff,⊙=5777K, and Δν≃α(νmax/μHz)β (here α=0.268μHz, β=0.758)[36]. fΔν is a typical asteroseismic correction factor, which is between 0.98 and 1.02 [37]. The predicted νmax values are calculated using the scaling relation [38]

    νmaxνmax⊙=(gg⊙)(TeffTeff,⊙)−0.5.

    		(9)

    where g is the surface gravity of the star and g⊙=27487cm/s2.

    B The magnetic monopole RC effect model in white dwarfs

    According to Eqs. (5)–(9), the number of magnetic monopoles captured from space and the total luminosity due to the RC effect by the MMs are given by

    Nm=7.18×1011nB(Rr∗)5ϕ(t9Mr∗)[1+4.256v−2−3(Mr∗Rr∗)].

    		(10)

    Lm≈4π3r3cnmnB⟨σmvT⟩mBc2=NmnB⟨σmvT⟩mBc2,=1.15×109n2B(Rr∗)5ϕt9(Mr∗)−1×⟨σmvT⟩[1+4.256v−2−3(Mr∗Rr∗)].

    		(11)

    Defining ξ=ϕ⟨σmvT⟩−28=ϕ⟨σmvT⟩/10−28, Eq. (11) may be rewritten

    Lm≈4π3r3cnmnB⟨σmvT⟩mBc2=NmnB⟨σmvT⟩mBc2,=1.15×1037n2B(Rr∗)5t9(Mr∗)−1ξ[1+4.256v−2−3(Mr∗Rr∗)].

    		(12)

    From Eqs. (1) and (12), and Lm=Lrad, we obtain

    ξ=5.29×10−15σMr∗t−19n−2B(Rr∗)−3T4eff[1+4.256v−2−3(Mr∗Rr∗)]−1.

    		(13)

    IV. RESULTS AND DISCUSSIONS

    The study of MMs has held considerable interest since MMs were found to be a generic feature of grand unified gauge theories in the physical fields. The theoretical predictions of monopole abundance are problematic in the standard cosmology, as far too many monopoles would have survived annihilation for the universe to have reached its present state. For example, the galactic field that yields the Parker bound is ϕ(σv)−28≤10−16cm−2s−1sr−1 [39]. Due to MM RC decay, another limit of the flux may be ϕ(σv)−28≤10−21cm−2s−1sr−1[31]. A better-understood limit in WDs may be ϕ(σv)−28≤10−18cm−2s−1sr−1 [32]. Then the bound has been stated as ϕ(σv)−28≤10−28cm−2s−1sr−1 by Freese and Krasteva [40]. In this paper, we study MMs and their numbers in space and discuss our MM model and the luminosity due to RC effect. We select the following typical parameters: m=1015,1017GeV, σm=10−24cm2, ϕ=5.59×10−28,10−26,7.59×10−26cm−2s−1sr−1, and nB=5.99×1031,1.89×1032cm−3, t9=1,10.

    Everyone knows that WDs evolve from red giants with mass less than 8 M⊙, we focus on 13 typical RGB stars listed in Tables 1 and 2 [41] and discuss their mass, radius, and cooling age. Depending on an asteroseismic correction factor, we propose five models (I−V) to examine the problem of the energy source of WDs.

    Table 1

    Table 1.  Information of the 13 red giant stars selected [41].
    starTIC IDGaia IDGania magDistance (pc)Galactic substructureSpectral source
    1TIC20897763Gaia 23656494710338280969.41484457.879 ± 9.435Gaia Enceladus SausageAPOGEE
    2TIC341816936Gaia 142177604633572300811.6231547.85 ± 45.63Gaia Enceladus SausageAPOGEE
    3TIC393961551Gaia 15063876279179368969.57166500.533 ± 6.312Gaia Enceladus SausageAPOGEE
    4TIC453888381Gaia 523025673034745715210.9714788.614 ± 15.999HaloGALAH
    5TIC279510617Gaia 548055045064301721610.7551933.263 ± 22.520HaloGALAH
    6TIC300938910Gaia 527067501829784422410.5629607.156 ± 7.7075HaloGALAH
    7TIC198204598Gaia 162989868534727385610.9455952.885 ± 37.512Halo···
    8TIC1008989Gaia 37896392809526103049.72882370.56 ± 5.85Gaia Enceladus Sausage···
    9TIC91556382Gaia 506500965033314739210.0855870.289 ± 34.565Halo···
    10TIC159509702Gaia 170919509028171827212.15421595.67 ± 55.605Halo···
    11TIC25079002Gaia 46693160657002229769.91465716.804 ± 15.5995DiskAPOGEE
    12TIC177242602Gaia 526229539536721228810.1451532.831 ± 14.294DiskAPOGEE
    13TIC9113677Gaia 324548565060765158410.1791491.438 ± 10.040Thick Disk···
    DownLoad: CSV
    Show Table

    Table 2

    Table 2.  Information of the 13 RGB stars [41].
    starνmaxΔν /μHzTeff /K[Fe/H][α/Fe]
    161.31383 ± 1.217686.81739 ± 0.249904988 ± 127−1.274 ± 0.0190.219 ± 0.021
    236.34 ± 0.764.2969 ± 0.07155068 ± 100−1.873 ± 0.1070.248 ± 0.023
    361.34 ± 1.756.68 ± 0.415121 ± 105−1.0751 ± 0.01230.156 ± 0.014
    450.37 ± 1.595.953 ± 0.0494741 ± 100−0.728 ± 0.070.32 ± 0.021
    528.57 ± 0.163.566 ± 0.0154450 ± 100−0.49 ± 0.050.281 ± 0.017
    6106.30 ± 0.929.464 ± 0.1314908 ± 100−0.792 ± 0.050.2566 ± 0.0165
    745.86 ± 0.315.132 ± 0.0324979 ± 100······
    8104.33059 ± 1.466189.80317 ± 0.153364893 ± 100······
    931.38665 ± 1.030974.189 ± 0.1865192 ± 100······
    1045.37 ± 0.535.090 ± 0.0274724 ± 100······
    1145.238 ± 0.624.967 ± 0.1214797 ± 830.1636 ± 0.006−0.010 ± 0.006
    1254.66 ± 0.335.663 ± 0.0314603 ± 100−0.1176 ± 0.0060.125 ± 0.007
    1387.22437 ± 0.870428.365 ± 0.1984764 ± 100······
    DownLoad: CSV
    Show Table

    In this paper, we select the asteroseismic correction factors f△ν= 0.98, 0.99, 1.00, 1.01, 1.02 for our typical models under study, which correspond to models (I − V). Based on Eqs. (8) and (9), we can calculate the mass and the radii of RGB stars Mr∗ (I − V) and Rr∗ (I − V) given in Table 3. We determine the stellar (RGB) ages using the packages BASTA [42], isochrones [43], isoclassify [44], PARAM [45, 46], and scaling-giants [47]. These ages correspond to t9 (I ∼ V) in Table 4.

    Table 3

    Table 3.  The mass and radii of the 13 RGB stars selected [41]. The asteroseismic correction factor f△ν= 0.98, 0.99, 1.00, 1.01, 1.02 defines models I − V, respectively.
    starM∗(I)M∗(II)M∗(III)M∗(IV)M∗(V)R∗(I)R∗(II)R∗(III)R∗(IV)R∗(V)
    10.89160.92860.96671.00591.04646.95417.09677.24087.38637.5333
    21.20471.25461.30611.35911.413810.45810.67210.88911.10811.329
    31.00751.04931.09231.13671.18247.34217.49277.64497.79857.9537
    40.78790.82060.85420.88890.92467.30457.45437.60567.75857.9129
    51.01541.05751.10091.14561.191611.18611.41611.64711.88212.118
    61.22111.27171.32391.37761.43306.20576.33306.46166.59146.7226
    71.15871.20671.25621.30721.35989.17039.35849.54849.74049.9342
    80.98961.04041.08301.12701.17235.66995.78625.90376.02246.1422
    90.89100.92800.96601.00521.04579.61939.816610.01610.21710.421
    100.92210.96030.99971.04031.08218.54488.72018.89729.07609.2566
    111.19861.24831.29951.35231.40669.47889.67329.869610.06810.268
    121.17621.22491.27521.32701.38038.63088.80788.98679.16739.3497
    131.05871.10261.14781.19451.24256.42506.55686.68996.82446.9602
    DownLoad: CSV
    Show Table

    Table 4

    Table 4.  The cooling age of the 13 RGB stars selected [41]. The five cooling ages correspond to models I − V. The discussions of the cooling ages are based on the packages BASTA [42], isochrones [43], isoclassify [44], PARAM [45, 46], and scaling-giants [47].
    starscaling-giants Age (Gyr)isochrones Age (Gyr)isoclassify Age (Gyr)PARAM Age (Gyr)BASTA Age (Gyr)
    t9(I)t9(II)t9(III)t9(IV)t9(V)
    15.8 ± 3.08.77 ± 2.95.68+0.79−0.839.29+2.82−3.239.0+2.7−2.5
    22.9 ± 1.89.16 ± 2.757.44+1.36−1.346.52+4.46−3.807.9+2.8−1.9
    34.7 ± 6.05.93 ± 3.055.63+0.84−0.695.72+4.61−3.147.5+3.3−2.2
    410.5 ± 3.59.72 ± 2.5012.78+0.78−1.5911.66+1.50−2.4214.8±2.8
    56.4 ±1.17.99 ± 3.3710.23+0.95−2.347.68+4.55−4.457.6+0.9−0.8
    62.7 ± 0.83.693 ± 0.7023.34+0.40−0.332.11+0.61−0.492.9+0.6−0.4
    73.3 ± 0.88.749 ± 2.1611.54+0.66−2.831.98+7.13−0.775.8+1.6−1.2
    85.8 ± 2.06.843 ± 3.336.84+1.50−0.907.27+2.95−2.0910.6+3.5−2.9
    97.1 ± 5.06.841 ± 3.596.43+1.89−0.965.27+2.84−1.778.9+3.8−2.9
    104.8 ± 1.29.344 ± 2.3319.98+1.47−1.187.92+3.47−2.4910.0+3.2−2.5
    112.1 ± 3.27.07 ± 3.651.92+0.17−0.111.84+0.18−0.143.6+0.3−0.4
    124.4 ± 2.77.49 ± 3.456.54+0.44−0.647.60+4.02−3.746.4+0.6−1.0
    135.1±2.07.07 ± 3.656.77+1.9−1.358.24+3.24−2.879.7+3.3−2.3
    DownLoad: CSV
    Show Table

    Based on Eqs. (7)–(9), there are two global seismic parameters, the frequency of maximum power, νmax, and the mean large frequency separation, △ν , to describe the oscillations of solar-like stars and radius. The surface gravity and temperature strongly determine the value of the frequency of maximum power, which is given by νmax∝gT−0.5eff∝MR−2T−0.5eff [48]. On the other hand, the travel time of sound from the center to the surface of a star will directly influence △ν, which is sensitive to the mean stellar density and is given by △ν∝ρ0.5∝M0.5R−1.5 [49]. In order to reduce systematic errors, by considering the effect on mass, [Fe/H] and Teff interpolation over a grid of models, a modification strategy was adopted by Sharma et al. [37]. Therefore, the asteroseismic correction factor in this paper is selected as f△ν= 0.98, 0.99, 1.00, 1.02, we propose five model (I−V). Detailed information on the mass and radius of RGB stars is shown in Table 3.

    Based on Eqs. (10) and (12), the age of RGB stars is a key parameter for estimating the number of MMs captured and the luminosity of WDs due to the RC effect by MMs. By combining masses inferred from the asteroseismic parameters with stellar atmospheric parameters and using stellar isochrones, the cooling ages of RGB stars were determined as shown in Table 4 by t9. Using the packages BASTA [42], isochrones [43], isoclassify [44], PARAM [45, 46], and scaling-giants [47], estimates are obtained of the cooling ages of RGB stars, which are the stellar ages since zero-age main sequence stars are considered.

    The scaling-giants package accepts asteroseismic parameters, metallicity, and temperature as inputs for model I. In the isochrones, isoclassify, PARAM, and BASTA packages, we take asteroseismic, photometric, and spectroscopic parameters as inputs for models II−V. From the SYD pipeline [50], along with effective temperatures determined through the direct method of isoclassification and metallicity determined by either the APOGEE or GALAH surveys, and using measured seismic νmax and △ν values, we can determine the ages with scaling-giants for model I. The SYD pipeline is an automated pipeline to estimate global oscillation parameters, such as the frequency of maximum power (νmax) and the large frequency spacing (△ν), for a large number of time series. By using 2MASS K magnitudes, asteroseismic △ν, Gaia parallaxes, and temperatures and metallicities from spectroscopy, we can determine the ages with isochrones, isoclassify, and PARAM, and BASTA packages for models II−V, respectively. Some useful parameters are presented in Tables 1–3. Detailed discussions are given in Grunblatt et al. [41].

    Based on Eq. (10), we give a computational assessment for the number of MMs captured for WDs in Tables 5 and 6 for m=1015,1017GeV, respectively. Our results in Table 5 show that the maximum number of MMs captured in models I ∼V are 4.8126×1022 and 6.5345×1024 when ϕ=5.59×10−28cm−2s−1sr−1, and ϕ=7.59×10−26cm−2s−1sr−1, respectively. Table 6 also shows that the number of MMs captured has a maximum of 9.1223×1024 (e.g., when m=1017GeV,nB=5.99×10−31cm−3,ϕ=7.59×10−26cm−2s−1sr−1, for model III). One can see that the number of MMs captured increases as the flux of MMs increases due to Nm∝ϕ according to Eq. (10). On the other hand, when the flux and mass of MMs are certain, there is no significant difference found in the number of MMs captured for same stars among the five models. However, for different models, there is a small difference. The reasons for these differences can be from the differences of the cooling ages for RGB stars due to differences in the stellar parameters selected, such as the asteroseismic correction factor and [Fe/H]. From the above analysis, it may be seen that the number of MMs captured can reach the maximum value of 9.1223×1024 when m=1017GeV,nB=5.99×10−31cm−3,ϕ=7.59×10−26cm−2s−1sr−1.

    Table 5

    Table 5.  The number of MMs captured in the five models I−V (corresponding to typical asteroseismic correction factor values f△ν=0.98,0.99,1.00,1.01,1.02 ) when m=1015GeV,nB=5.99×1031cm−3,ϕ=5.59×10−28,7.59×10−26cm−2s−1sr−1.
    ϕ=5.59×10−28cm−2s−1sr−1ϕ=7.59×10−26cm−2s−1sr−1
    starNm(I)Nm(II)Nm(III)Nm(IV)Nm(V)Nm(I)Nm(II)Nm(III)Nm(IV)Nm(V)
    12.5599e214.1144e212.8307e214.9152e215.0524e213.4758e235.5864e233.8435e236.6738e236.8601e23
    27.2823e212.4449e222.7645e221.9626e222.5231e229.8877e233.3197e243.7536e242.6648e243.4258e24
    32.4095e213.2314e213.2591e213.5154e214.8906e213.2716e234.3876e234.4251e234.7731e236.6404e23
    46.6993e216.5919e219.2068e218.9176e211.2010e229.0962e238.9503e231.2501e241.2108e241.6307e24
    52.6667e223.5386e224.8126e223.8356e224.0272e223.6207e244.8046e246.5345e245.2079e245.4680e24
    64.9402e207.1828e206.9014e204.6290e206.7508e206.7077e229.7527e229.3705e226.2851e229.1661e22
    74.4692e211.2594e221.7647e221.4790e229.9908e216.0681e231.7100e242.3961e242.0081e241.3565e24
    88.2517e201.0349e211.0989e211.2401e211.9185e211.1204e231.4051e231.4921e231.6837e232.6049e23
    91.5855e221.6237e221.6212e221.4106e222.5275e222.1527e242.2046e242.2012e241.9152e243.4318e24
    105.7326e211.1862e221.3458e221.1338e221.5189e227.7836e231.6105e241.8273e241.5395e242.0624e24
    113.2438e211.1608e223.3488e213.4071e217.0729e214.4044e231.5761e244.5469e234.6261e239.6034e23
    124.3369e217.8473e217.2788e218.9801e218.0238e215.8885e231.0655e249.8830e231.2193e241.0895e24
    131.2784e211.8839e211.9164e212.4764e213.0932e211.7359e232.5579e232.6020e233.3624e234.1999e23
    DownLoad: CSV
    Show Table

    Table 6

    Table 6.  The number of MMs captured in the five models I−V (corresponding to typical asteroseismic correction factor values fΔν=0.98,0.99,1.00,1.01,1.02 ) when m=1017GeV,nB=5.99×1031cm−3,ϕ=5.59×10−28,7.59×10−26cm−2s−1sr−1.
    ϕ=5.59×10−28cm−2s−1sr−1ϕ=7.59×10−26cm−2s−1sr−1
    starNm(I)Nm(II)Nm(III)Nm(IV)Nm(V)Nm(I)Nm(II)Nm(III)Nm(IV)Nm(V)
    13.9330e216.3662e214.4113e217.7145e217.9867e215.3401e238.6439e235.9895e231.0475e241.0844e24
    21.0794e223.6479e224.1522e222.9675e223.8405e221.4656e244.9531e245.6377e244.0292e245.2146e24
    33.7921e215.1235e215.2058e215.6570e217.9291e215.1488e236.9565e237.0683e237.6810e231.0766e24
    49.7253e219.6300e211.3536e221.3195e221.7884e221.3205e241.3075e241.8379e241.7915e242.4282e24
    53.6811e224.9121e226.7186e225.3850e225.6862e224.9981e246.6696e249.1223e247.3117e247.7206e24
    68.9937e201.3196e211.2795e218.6609e201.2747e211.2212e231.7918e231.7373e231.1760e231.7307e23
    76.8316e211.9388e222.7358e222.3091e221.5709e229.2759e232.6324e243.7146e243.1352e242.1330e24
    81.4320e211.8114e211.9401e212.2082e213.4458e211.9443e232.4595e232.6342e232.9982e234.6787e23
    92.2008e222.2668e222.2762e221.9920e223.5898e222.9883e243.0778e243.0906e242.7046e244.8742e24
    108.3230e211.7331e221.9789e221.6778e222.2622e221.1301e242.3532e242.6868e242.2781e243.0715e24
    114.9599e211.7875e225.1930e215.3209e211.1124e226.7345e232.4270e247.0510e237.2247e231.5105e24
    126.8084e211.2410e221.1597e221.4414e221.2975e229.2444e231.6851e241.5746e241.9571e241.7617e24
    132.1582e213.2067e213.2891e214.2856e215.3977e212.9304e234.3540e234.4659e235.8190e237.3289e23
    DownLoad: CSV
    Show Table

    Figures 1 and 2 display the luminosities as a function of t9 of WDs of the five models under different astronomical conditions. From the two figures, it can be seen that the calculated luminosities agree well with observations. One can also conclude the same from Table 7. For example, the ranges of our calculated luminosities are 1.3336×1034∼7.1985×1036ergs−1 and 1.8224×1034∼1.0871×1036ergs−1 for models I and V, respectively, in Table 7.

    Figure 1

    Figure 1.  (color online) The luminosities as a function of t9 for the 13 WDs [41] for the five models I −V when nB=5.99×1031cm−3, m=1015GeV, σm=10−24cm2,ϕ=10−26cm−2s−1sr−1 at the temperatureT6=1.
    DownLoad: Full-Size Img PowerPoint

    Figure 2

    Figure 2.  (color online) The luminosities as a function of t9 of the 13 WDs [41] for models I − V when nB=5.99×1031cm−3, m=1015GeV, σm=10−24cm2,ϕ=10−26cm−2s−1sr−1 at the temperature T6=10.
    DownLoad: Full-Size Img PowerPoint

    Table 7

    Table 7.  Comparisons of the calculated luminosities (Lm) due to the RC effect by MMs with observations (Lrad) for the five models I − V at nB=5.99×1031cm−3,m=1015GeV, σm=10−24cm2,ϕ=10−26cm−2s−1sr−1,T6=1 (ξ=1.5745×10−15cm−2s−1sr−1).
    starour resultsobservations
    Lm(I)Lm(II)Lm(III)Lm(IV)Lm(V)Lrad(I)Lrad(II)Lrad(III)Lrad(IV)Lm(V)
    16.9103e341.1107e357.6414e341.3269e351.3639e351.0318e351.0746e351.1187e351.1641e351.2109e35
    21.9658e356.6000e357.4626e355.2980e356.8110e352.4869e352.5900e352.6962e352.8057e352.9185e35
    36.5043e348.7231e348.7978e349.4896e341.3202e351.2778e351.3308e351.3854e351.4417e351.4996e35
    41.8085e351.7795e352.4853e352.4073e353.2420e359.2913e349.6763e341.0073e351.0482e351.0904e35
    57.1985e359.5522e351.2992e361.0354e361.0871e361.6913e351.7614e351.8336e351.9081e351.9848e35
    61.3336e341.9390e341.8630e341.2496e341.8224e347.7022e348.0214e348.3505e348.6895e349.0388e34
    71.2064e353.3998e354.7637e353.9925e352.6970e351.7814e351.8552e351.9313e352.0097e352.0905e35
    82.2275e342.7936e342.9665e343.3475e345.1790e346.3515e346.6147e346.8861e347.1657e347.4537e34
    94.2799e354.3831e354.3763e353.8078e356.8228e352.3176e352.4137e352.5127e352.6147e352.7198e35
    101.5475e353.2020e353.6329e353.0607e354.1003e358.3978e348.7458e349.1046e349.4742e349.8551e34
    118.7565e343.1336e359.0399e349.1974e341.9093e351.6398e351.7078e351.7779e351.8501e351.9244e35
    121.1707e352.1183e351.9649e352.4242e352.1660e351.1526e351.2004e351.2496e351.3004e351.3526e35
    133.4511e345.0855e345.1732e346.6849e348.3500e347.3599e347.6649e347.9794e348.3034e348.6371e34
    DownLoad: CSV
    Show Table

    In order to facilitate comparison of our results with the observed data, we define the scale factor kias the ratio of our results due to the RC effect to the observed luminosity. We note that the largest differences between our results and the observed values are one order of magnitude. For example, based on Table 7 and Table 9 for star 5, the maximal scale factors are k1=4.2562, k2=5.4231, k3=7.0851, k4=5,4264, and k5=5.4772 for models I−V, respectively. However, in Table 8 and Table 9, the maximal scale factors for star 5 are k1=13.459, k2=17.149, k3=22.405, k4=17.160 and k5=17.321 for models I−V, respectively.

    Table 9

    Table 9.  The comparisons of the scale factor ki (i=(1−5) indexes models I−V), which is the ratio of our results (Lm) due to RC effect by MMs to the observed luminosities (Lrad) for the five models I−V from Tables 7 and 8.
    starTable 7Table 8
    k1k2k3k4k5k1k2k3k4k5
    10.669731.03360.683091.13981.12642.11793.26852.16013.60453.5619
    20.790482.54832.76781.88832.33382.49978.05858.75275.97147.3801
    30.50900.65550.63500.65820.88041.60962.07282.00822.08152.7840
    41.94641.83902.46732.29652.97336.15515.81537.80227.26239.4024
    54.25625.42317.08515.42645.477213.45917.14922.40517.16017.321
    60.17310.24170.22310.14380.20160.547530.76440.70550.45470.6376
    70.67731.83262.46661.98661.29012.14165.79517.80006.28214.0797
    80.35070.422330.43080.46720.69481.10901.33551.36231.47732.1972
    91.84671.81591.74171.45632.50855.83965.74255.50764.60527.9327
    101.84273.66123.99023.23064.16065.827311.57812.61810.21613.157
    110.53401.83480.50850.49710.99221.68865.80231.60791.57213.1374
    121.01571.76471.57241.86421.60133.21205.58064.97235.89525.0638
    130.46890.66350.64830.80510.96681.48282.09812.05022.54593.0572
    DownLoad: CSV
    Show Table

    Table 8

    Table 8.  The comparisons of the luminosity (Lm) due to RC effect by MMs with the observed values (Lrad) for the five models I − V at nB=5.99×1031cm−3,m=1015GeV, σm=10−24cm2,ϕ=10−26cm−2s−1sr−1,T6=10 (ξ=4.9790×10−15cm−2s−1sr−1).
    starour resultsobservations
    Lm(I)Lm(II)Lm(III)Lm(IV)Lm(V)Lrad(I)Lrad(II)Lrad(III)Lrad(IV)Lm(V)
    12.1852e353.5122e352.4164e354.1959e354.3130e351.0318e351.0746e351.1187e351.1641e351.2109e35
    26.2165e352.0871e362.3599e361.6754e362.1538e362.4869e352.5900e352.6962e352.8057e352.9185e35
    32.0568e352.7585e352.7821e353.0009e354.1749e351.2778e351.3308e351.3854e351.4417e351.4996e35
    45.7189e355.6271e357.8593e357.6125e351.0252e369.2913e349.6763e341.0073e351.0482e351.0904e35
    52.2764e363.0207e364.1083e363.2743e363.4378e361.6913e351.7614e351.8336e351.9081e351.9848e35
    64.2172e346.1316e345.8913e343.9515e345.7628e347.7022e348.0214e348.3505e348.6895e349.0388e34
    73.8151e351.0751e361.5064e361.2625e368.5286e351.7814e351.8552e351.9313e352.0097e352.0905e35
    87.0440e348.8342e349.3808e341.0586e351.6377e356.3515e346.6147e346.8861e347.1657e347.4537e34
    91.3534e361.3861e361.3839e361.2041e362.1576e362.3176e352.4137e352.5127e352.6147e352.7198e35
    104.8936e351.0126e361.1488e369.6788e351.2966e368.3978e348.7458e349.1046e349.4742e349.8551e34
    112.7691e359.9092e352.8587e352.9085e356.0377e351.6398e351.7078e351.7779e351.8501e351.9244e35
    123.7022e356.6988e356.2135e357.6659e356.8495e351.1526e351.2004e351.2496e351.3004e351.3526e35
    131.0913e351.6082e351.6359e352.1140e352.6405e357.3599e347.6649e347.9794e348.3034e348.6371e34
    DownLoad: CSV
    Show Table

    According to the analysis above, our results agree well with observations at lower relativistic densities and temperatures, but the greatest difference is about two orders of magnitude at higher relativistic densities and temperatures in WDs. The good agreement of our calculatedresults with observations shows that our model with the RC effect by MMs yields realistic results, as well as suggesting that the energy source of WDs is the RC effect by MMs.

    The monopole flux problem is well known to be a highly challenging and interesting issue. Some scholars have made pioneering work on this subject, such as Freese [32], Parker [39], Kolb & Turner [31], and Freese & Krasteva [40]. In this paper, we discuss this problem by considering the RC effect of MMs. Tables 10 (m=1015GeV,β=9.4868×10−3) and 11 (m=1017GeV,β=1.00×10−3) show the flux of MMs for the four models I∼III and V, which correspond to typical asteroseismic correction factor values f△ν=0.98,0.99,1.00,1.02 when nB=5.99×1031,1.89×1032cm−3. One can see that the maximum of the monopole flux values are 9.0935×10−13cm−2s−1sr−1 and 5.8519×10−13cm−2s−1sr−1 in Tables 10 and 11, respectively.

    Table 10

    Table 10.  The flux of MMs for the five models I−V (corresponding to typical asteroseismic correction factor values fΔν= 0.98, 0.99, 1.00, 1.01, 1.02) when m=1015GeV,β=9.4868×10−3,nB=5.99×1031,1.89×1032cm−3.
    starnB=5.99×1031cm−3nB=1.89×1032cm−3
    ξ(I)ξ(II)ξ(III)ξ(IV)ξ(V)ξ(I)ξ(II)ξ(III)ξ(IV)ξ(V)
    12.3509e-131.5233e-132.3050e-131.3813e-131.3978e-132.3614e-141.5301e-142.3152e-141.3875e-141.4041e-14
    21.9918e-136.1786e-145.6885e-148.3380e-146.7465e-142.0007e-146.2061e-155.7139e-158.3752e-156.7765e-15
    33.0933e-132.4021e-132.4794e-132.3920e-131.7884e-133.1070e-142.4128e-142.4904e-142.4026e-141.7964e-14
    48.0892e-148.5618e-146.3815e-146.8559e-145.2954e-148.1252e-158.5999e-156.4099e-156.8865e-155.3190e-15
    53.6993e-142.9033e-142.2223e-142.9015e-142.8746e-143.7157e-152.9162e-152.2322e-152.9145e-152.8874e-15
    69.0935e-136.5135e-137.0573e-136.0949e-137.8094e-139.1340e-146.5425e-147.0887e-147.0998e-147.8442e-14
    72.3248e-138.5916e-146.3833e-147.9257e-141.2204e-132.3352e-148.6299e-156.4117e-157.9610e-151.2259e-14
    84.4895e-133.7281e-133.6549e-133.3703e-132.2661e-134.5095e-143.7447e-143.6711e-143.3854e-142.2761e-14
    98.5262e-148.6703e-149.0402e-141.0812e-136.2765e-148.5642e-158.7090e-159.0804e-151.0860e-146.3044e-15
    108.5442e-144.3005e-143.9459e-144.8737e-143.7843e-148.5823e-154.3196e-153.9635e-154.8954e-153.8011e-15
    112.9486e-138.5810e-143.0965e-133.1671e-131.5870e-132.9617e-148.6192e-153.1103e-143.1812e-141.5940e-14
    121.5501e-138.9220e-141.0013e-138.4458e-149.8324e-141.5570e-148.9617e-151.0058e-148.4834e-159.8762e-15
    133.3578e-132.3731e-132.4285e-131.9557e-131.6286e-133.3727e-142.3837e-142.4394e-141.9644e-141.6359e-14
    DownLoad: CSV
    Show Table

    Table 11

    Table 11.  The flux of MMs for the five models I−V (corresponding to typical asteroseismic correction factor values fΔν= 0.98, 0.99, 1.00, 1.01, 1.02) when m=1017GeV,β=1.0×10−3,nB=5.99×1031,1.89×1032cm−3.
    starnB=5.99×1031cm−3nB=1.89×1032cm−3
    ξ(I)ξ(II)ξ(III)ξ(IV)ξ(V)ξ(I)ξ(II)ξ(III)ξ(IV)ξ(V)
    11.5302e-139.8450e-141.4791e-138.8011e-148.8428e-141.5370e-149.8889e-151.4857e-148.8403e-158.8822e-15
    21.3438e-134.1410e-143.7874e-145.5146e-144.4323e-141.3498e-144.1595e-153.8042e-155.5391e-154.4520e-15
    31.9655e-131.5150e-131.5522e-131.4864e-131.1031e-131.9742e-141.5218e-141.5591e-141.4930e-141.1080e-14
    45.5723e-145.8607e-144.3406e-144.6336e-143.5561e-145.5971e-155.8868e-154.3599e-154.6543e-153.5720e-15
    52.6798e-142.0914e-141.5918e-142.0667e-142.0359e-142.6918e-152.1008e-151.5989e-152.0759e-152.0450e-15
    64.9950e-133.5454e-133.8064e-135.8519e-134.1359e-135.0173e-143.5612e-143.8234e-145.8779e-144.1543e-14
    71.5209e-135.5812e-144.1175e-145.0764e-147.7618e-141.5276e-145.6060e-154.1359e-155.0990e-157.7963e-15
    82.5870e-132.1299e-132.0702e-131.8927e-131.2617e-132.5985e-142.1394e-142.0794e-141.9011e-141.2673e-14
    96.1421e-146.2105e-146.4385e-147.6561e-144.4190e-146.1695e-156.2382e-156.4672e-157.6902e-154.4387e-15
    105.8850e-142.9433e-142.6835e-143.2935e-142.5410e-145.9112e-152.9564e-152.6955e-153.3081e-152.5523e-15
    111.9284e-135.5727e-141.9968e-132.0279e-131.0090e-131.9370e-145.5975e-152.0057e-142.0370e-141.0135e-14
    129.8741e-145.6415e-146.2849e-145.2620e-146.0805e-149.9181e-155.6666e-156.3129e-155.2854e-156.1076e-15
    131.9890e-131.3942e-131.4150e-131.1301e-139.3331e-141.9979e-141.4004e-141.4213e-141.1351e-149.3746e-15
    DownLoad: CSV
    Show Table

    It is very interesting to note that the monopole flux decreases as nB increases from 5.99×1031 to 1.89×1032cm−3 in Tables 10 and 11. This is not hard to understand according to Eq. (10). Based on our calculations above from Tables 10, due to the RC effect by MMs, we obtain new limits on the MM flux of ξ≤9.0935×10−13cm−2s−1sr−1 and ξ≤9.1340×10−14cm−2s−1sr−1 when nB=5.99×1031cm−3 and 1.899×1032cm−3, respectively. In Table 11, the new limits on the MM flux are ξ≤4.9950×10−13cm−2s−1sr−1 and ξ≤5.0173×10−14cm−2s−1sr−1 when nB=5.99×1031 and 1.899×1032, respectively.

    Based on the above analysis, we obtain new limits on the MM flux of ξ≤9.0935×10−13cm−2s−1sr−1, and ξ≤4.9950×10−13cm−2s−1sr−1 at nB=5.99×1031cm−3 when m=1015GeV,β=9.4868×10−3, and m=1017GeV,β=10−3, respectively. When we estimate the number of MMs captured, the MM luminosity and the limit of the MM flux, as samples, we consider the 13 RGB stars in our MM model for the following reasons. First, WDs originate from RGB stars. Second, compared with WDs, RGB stars have a very large surface area. According to Eq. (10), we expect RGB phases to capture more MMs during their evolution period. Third, since the MM is a superheavy particle, when MMs are captured by an RGB, they will be deposited in the star core. If all MMs captured by RGBs remain, their number will be much larger than that of those captured by WDs. Thus, the number of MMs calculated inside an WD will be more accurate than for MMs captured only during the WD phase. For example, Freese et al. [51] showed that if the MMs captured by stars in the main sequence stage all survive, the MM flow due to neutron star catalysis can be strengthened by up to 7 orders of magnitude. Finally, based on Schwarzschid [52], the nuclear energy generation rates of the proton-proton and CNO cycle are ϵpp≈10ρ100T47ergg−1s−1and ϵCNO≈8ρ100T167ergg−1s−1, respectively (where T7=T/107K is the temperature, and ρ100=ρ/100 is the density). Based on the discussions fof Bjork et al. [53], we may select the mass of the outer layer of the RGB as being from 0.005∼0.02M⊙ (the main component is hydrogen). Thus, when T7=0.1−1, ρ100∼10−4, and we obtain a proton-proton nuclear energy generation rate of 1024−1028ergss−1, which is ≪Lm=1034−1036ergss−1 in our calculations. Based on the above analysis, and the fact that RGB stars are the origins of WDs, we therefore have Lm≈Lrad.

    One can also conclude that with the increasing number of MMs captured, the luminosity due to the RC effect by MMs increases linearly with time until it becomes the main contribution to the total luminosity. One can even observe that for some of the oldest white dwarfs, the luminosity may have passed its minimum and some reheating may have occurred.

    It may be suggested that the annihilation of magnetic and antimagnetic monopoles could result in a significant reduction in the number of monopoles and the catalytic luminosity of the monopoles in the WDs. Dicus et al. [54] calculated the annihilation cross sections of magnetic monopoles and antimonopoles caused by two-body and three-body recombination. Their results show that this kind of annihilation has little effect on the flux and luminosity of the monopole. On the other hand, some WDs may have magnetic fields of up to 105G according to observations. The forces generated by the magnetic field inside the white dwarf must balance the gravitational and Coulomb interactions.The magnetic field may keep the monopole and antimonopole distributions far enough apart for annihilation to be negligible.

    On the other hand, neutrino emission in WDs is a very interesting issue. Based on the discussions of Althaus et al. [55], when WD is very hot, neutrino emissions could be a major source of cooling. However, based on the relatively low temperature environment of WDs (e.g., T6=1,10 in our paper) we discussed the heating resource problem with our MMs model. The neutrino processes inside WDs at such low temperatures (e.g., T6=1) may not be the main cooling process (see discussions by Itoh et al. [56]). On the other hand, Izawa [57] also discussed the neutrinos emitted according to Eqs. (23), (24) in his paper and calculated the neutrino emitted per one nucleon decay at low and high energy components, finding that these neutrino losses did not affect the structure or evolution of Rubakov stars because the energy lost through the neutrino emission is smaller than 100 MeV per one nucleon decayn although about two neutrinos are emitted furing the decay of one nucleon.

    According to the above analysis, one can see that MMs pass through space to be captured by WDs. MMs trapped inside a WD can catalyze the decay of nuclei, which can function as an energy source to keep the WDs hot.

    V. CONCLUSIONS AND OUTLOOKS

    We have presented five MMs models of WD energy resources to discuss their cooling based on certain observations of 13 RGB stars. We find that the number of MMs captured can reach a maximum value of 9.1223×1024 when m=1017GeV,nB=5.99×1031cm−3,ϕ=7.59×10−26cm−2s−1sr−1. The good agreement with observations of our luminosities due to the RC effect by MMs calculated for WDs shows that our model is reasonable. We conclude that the energy source of WDs may be the RC effect. Due to the RC effect by MMs, we obtain a new limit of MM flux of ξ≤9.0935×10−13cm−2s−1sr−1 and ξ≤4.9950×10−13cm−2s−1sr−1 at nB=5.99×1031cm−3 when m=1015GeV,β=9.4868×10−3, and m=1017GeV,β=10−3, respectively.

    In this paper, the main highlights may be given as follows. First, we created detailed estimates of the cooling ages of 13 RGB stars using the packages BASTA [42], isochrones [43], isoclassify [44], PARAM [45,46], and scaling-giants [47]. Second, we proposes five new models to discuss the energy resources and the cooling of WDs and compare the luminosities with observations for 13 RGB stars due to the RC effect. Finally, the new limit of the MM flux is obtained based on our models.

    As is widely known, research on MMs haa always been a hot frontier topic in the fields of nuclear physics and astrophysics. The search for MMs remians a difficult and challenging problem, and the flux of magnetic monopoles in the universe remains uncertain. The neutrino emissivity rates due to the RC effect also may play a key role in the process of WD and neutron star evolution. These challenging problem will be our future issues.

    References

    [1] L. Mestel, MNRAS. 112, 583 (1952) doi: 10.1093/mnras/112.6.583
    [2] R. M. Avakian, CoMBAO. 44, 115 (1972) doi: https://ui.adsabs.harvard.edu/abs/1972CoBAO..44..115A/abstract
    [3] L. Bildsten and D. M. Hall, Astrophys. J. 549, 219 (2001) doi: 10.1086/319169
    [4] C. J. Deloye and L. Bildsten, Astrophys. J. 580, 1077 (2002) doi: 10.1086/343800
    [5] E. García-Berro, L. G. Althaus, A. H. Córsico et al., Asstrophys. J. 677, 473 (2008) doi: 10.1086/527536
    [6] R. V. Lobato, M. Malheiro, and J. G. Coelho, Mgm. Conf. 4313, 4318 (2018) doi: 10.1142/9789813226609_0578
    [7] S. H. Cheng, J. D. Cummings, and B. Ménard, Asstrophys. J. 886, 100 (2019) doi: 10.3847/1538-4357/ab4989
    [8] M. E. Caplan, C. J. Horowitz, and A. Cumming, Astrophys. J. 902(2), 44C (2020) doi: 10.3847/2041-8213/abbda0
    [9] C. Callan et al., Nucl. Phys. 212, 391 (1983) doi: 10.1016/0550-3213(83)90677-6
    [10] V. A. Rubakov, JETP Lett. 33, 644 (1981) doi: https://ui.adsabs.harvard.edu/abs/1981JETPL..33..644R/abstract
    [11] M. Detrixhe, D. Besson, P. W. Gorham et al., Phys. Lett. D 83, 023513 (2011) doi: https://journals.aps.org/prd/abstract/10.1103/PhysRevD.83.023513
    [12] M. Frank, A. Antoshkin, C. Dukes et al., ICRC 36, 888 (2019) doi: 10.22323/1.358.0888
    [13] T. Fujii and A. C. Pierre, ICRC 34, 319 (2015) doi: 10.22323/1.236.0319
    [14] B. Kain, Phys. Rev. D 100, 063003 (2019) doi: https://journals.aps.org/prd/abstract/10.1103/PhysRevD.100.063003
    [15] A. Rajantie, Phil. Trans. R. Soc. A. 370, 5705 (2012) doi: 10.1098/rsta.2011.0394
    [16] N. E. Mavromatos and V. A. Mitsou, IJMPA 35, 2030012 (2020) doi: 10.1142/S0217751X20300124
    [17] J. J. Liu, MNRAS. 433, 1108 (2013) doi: 10.1093/mnras/stt792
    [18] J. J. Liu, MNRAS. 438, 930 (2014) doi: 10.1093/mnras/stt2086
    [19] J. J. Liu and W. M. Gu, Astrophys. J. Suppl. 224, 29 (2016) doi: 10.3847/0067-0049/224/2/29
    [20] J. J. Liu, D. M. Liu et al., Chin. Phys. C 41(12), 125102 (2017) doi: 10.1088/1674-1137/41/12/125102
    [21] J. J. Liu and D. M. Liu, EPJC. 78, 84 (2018a) doi: 10.1140/epjc/s10052-018-5559-9
    [22] J. J. Liu and D. M. Liu, RAA. 18(1), 8 (2018) doi: 10.1088/1674–4527/18/1/8
    [23] J. J. Liu and D. M. Liu, Astrophys. Space Sci. 363(5), 115 (2018) doi: 10.1007/s10509-018-3331-0
    [24] J. J. Liu and D. M. Liu, New Astron. 69, 69 (2019a) doi: 10.1016/j.newast.2018.12.003
    [25] J. J. Liu, D. M. Liu, and L. H. Hao, ChPhC 43, 064107 (2019b) doi: https://iopscience.iop.org/article/10.1088/1674-1137/43/6/064107
    [26] J. J. Liu and D. M. Liu, Astron. Nachr. 341, 291 (2020) doi: https://onlinelibrary.wiley.com/doi/full/10.1002/asna.202013537
    [27] Q. H., Peng, J. J. Liu, and C. K. Chou, MPLA 35, 2050030 (2020) doi: 10.1142/S0217732320500303
    [28] Q. H. Peng, J. J. Liu, and Z. Q. Ma, New Astron. 57, 59 (2017) doi: 10.1016/j.newast.2017.06.011
    [29] W. Bernreuther and N. S. Craigie, Phys. Rev. Lett. 55, 2555 (1985) doi: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.55.2555
    [30] Q. H. Peng, Z. Y. Lie, and D. Y. Wang, Sci. Sin. Phys. Mech. As. 28, 970 (1985) doi: https://ui.adsabs.harvard.edu/abs/1985SSSMP..28..970P/abstract
    [31] E. Kolb and M. S. Turner., Astrophys. J. 286, 702 (1984) doi: 10.1086/162645
    [32] K. Freese, Asstrophys. J. 286, 216 (1984) doi: 10.1086/162589
    [33] Z. Q. Ma and J. F. Tang, Phys. Lett. B 153, 59 (1983) doi: https://linkinghub.elsevier.com/retrieve/pii/0370269383908158
    [34] T. Kallinger, W. W. Weiss, C. Barban et al., A&A 509, A77 (2010) doi: https://www.aanda.org/articles/aa/full_html/2010/01/aa11437-08/aa11437-08.html
    [35] D. Stello, and H. Bruntt, H. Preston et al., Astrophys. J. 674, L53 (2008) doi: 10.1086/528936
    [36] D. Huber, T. R. Bedding, D. Stello et al., Astrophys. J. 743, 143 (2011) doi: 10.1088/0004-637X/743/2/143
    [37] S. Sharma, D. Stello, and J. Bland-Hawthorn et al., Astrophys. J. 822, 15 (2016) doi: 10.3847/0004-637X/822/1/15
    [38] T. M. Brown, D. W. Latham, M. E. Everett et al., Astron. J. 142, 112 (2011) doi: 10.1088/0004-6256/142/4/112
    [39] E. Parker, Asstrophys. J. 160, 383 (1970) doi: 10.1086/150442
    [40] K. Freese and E. Krasteva, Phys. Rev. D 59, 063007 (1999) doi: https://journals.aps.org/prd/abstract/10.1103/PhysRevD.59.063007
    [41] S. K. Grunblatt, J. C. Zinn, A. M. Price-Whelan et al., Astrophys. J. 916, 88 (2021) doi: 10.3847/1538-4357/ac0532
    [42] V. Silva Aguirre, G. R. Davies, and S. Basu, MNRAS 452, 2127 (2015) doi: 10.1093/mnras/stv1388
    [43] T. D. Morton, Isochrones: Stellar Model Grid Package, v2.0, (Astrophysics Source Code Library, ascl: 1503.010, 2015)
    [44] D. Huber, J. Zinn, M. Bojsen-Hansen et al., Astrophys. J. 844, 102 (2017) doi: 10.3847/1538-4357/aa75ca
    [45] L. da Silva, L. Girardi, L. Pasquini et al., A&A 458, 609 (2006) doi: https://www.aanda.org/htbin/resolve?bibcode=2006A%26A...458..609DFUL
    [46] T. S. Rodrigues, D. Bossini, and A. Miglio, Mon. Not. R. Astron. Soc. 467, 1433 (2017) doi: https://academic.oup.com/mnras/article/467/2/1433/2917991
    [47] E. P. Bellinger, Mon. Not. R. Astron. Soc. 492, L50 (2020) doi: 10.1093/mnrasl/slz178
    [48] K. Belkacem, M. J. Goupil, M. A. Dupret et al., A&A 530, A142 (2011) doi: 10.1051/0004-6361/201116490
    [49] H. Kjeldsen and T. R. Bedding, A&A 293, 87 (1995)
    [50] D. Huber, D. Stello, T. R. Bedding et al., Comm. Asteroseismology 160, 74 (2009) doi: https://articles.adsabs.harvard.edu/pdf/2009CoAst.160...74H
    [51] K. Freese, M. S.Turner, and D. N. Schramm, Phys. Rev. Lett. 51, 1625 (1983) doi: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.51.1625
    [52] M. Schwarzschild, Structure and Evolution of the Star., (Princeton: Princeton university Press, 1958)
    [53] R. Bjork, Stephan, and B. Chaboyer, Astrophys. J. 641, 1102 (2006) doi: 10.1086/500505
    [54] D. A. Dicus, D. N. Page, and V. L. Teplitz, Phys. Rev. D 26, 1306 (1982) doi: https://journals.aps.org/prd/abstract/10.1103/PhysRevD.26.1306
    [55] L. G. Althaus, A. H. Córsico, J. Isern et al., Astron. Astrophys. Rev. 18, 471 (2010) doi: https://link.springer.com/article/10.1007/s00159-010-0033-1
    [56] N. Itoh, H. Hayashi, A. Nishikawa et al., Astrophys. J. Suppl. 102, 411 (1996) doi: 10.1086/192264
    [57] M. Izawa, Prog. Theor. 75, 556 (1986) doi: https://academic.oup.com/ptp/article/75/3/556/1915730

  • References

    [1] L. Mestel, MNRAS. 112, 583 (1952) doi: 10.1093/mnras/112.6.583
    [2] R. M. Avakian, CoMBAO. 44, 115 (1972) doi: https://ui.adsabs.harvard.edu/abs/1972CoBAO..44..115A/abstract
    [3] L. Bildsten and D. M. Hall, Astrophys. J. 549, 219 (2001) doi: 10.1086/319169
    [4] C. J. Deloye and L. Bildsten, Astrophys. J. 580, 1077 (2002) doi: 10.1086/343800
    [5] E. García-Berro, L. G. Althaus, A. H. Córsico et al., Asstrophys. J. 677, 473 (2008) doi: 10.1086/527536
    [6] R. V. Lobato, M. Malheiro, and J. G. Coelho, Mgm. Conf. 4313, 4318 (2018) doi: 10.1142/9789813226609_0578
    [7] S. H. Cheng, J. D. Cummings, and B. Ménard, Asstrophys. J. 886, 100 (2019) doi: 10.3847/1538-4357/ab4989
    [8] M. E. Caplan, C. J. Horowitz, and A. Cumming, Astrophys. J. 902(2), 44C (2020) doi: 10.3847/2041-8213/abbda0
    [9] C. Callan et al., Nucl. Phys. 212, 391 (1983) doi: 10.1016/0550-3213(83)90677-6
    [10] V. A. Rubakov, JETP Lett. 33, 644 (1981) doi: https://ui.adsabs.harvard.edu/abs/1981JETPL..33..644R/abstract
    [11] M. Detrixhe, D. Besson, P. W. Gorham et al., Phys. Lett. D 83, 023513 (2011) doi: https://journals.aps.org/prd/abstract/10.1103/PhysRevD.83.023513
    [12] M. Frank, A. Antoshkin, C. Dukes et al., ICRC 36, 888 (2019) doi: 10.22323/1.358.0888
    [13] T. Fujii and A. C. Pierre, ICRC 34, 319 (2015) doi: 10.22323/1.236.0319
    [14] B. Kain, Phys. Rev. D 100, 063003 (2019) doi: https://journals.aps.org/prd/abstract/10.1103/PhysRevD.100.063003
    [15] A. Rajantie, Phil. Trans. R. Soc. A. 370, 5705 (2012) doi: 10.1098/rsta.2011.0394
    [16] N. E. Mavromatos and V. A. Mitsou, IJMPA 35, 2030012 (2020) doi: 10.1142/S0217751X20300124
    [17] J. J. Liu, MNRAS. 433, 1108 (2013) doi: 10.1093/mnras/stt792
    [18] J. J. Liu, MNRAS. 438, 930 (2014) doi: 10.1093/mnras/stt2086
    [19] J. J. Liu and W. M. Gu, Astrophys. J. Suppl. 224, 29 (2016) doi: 10.3847/0067-0049/224/2/29
    [20] J. J. Liu, D. M. Liu et al., Chin. Phys. C 41(12), 125102 (2017) doi: 10.1088/1674-1137/41/12/125102
    [21] J. J. Liu and D. M. Liu, EPJC. 78, 84 (2018a) doi: 10.1140/epjc/s10052-018-5559-9
    [22] J. J. Liu and D. M. Liu, RAA. 18(1), 8 (2018) doi: 10.1088/1674–4527/18/1/8
    [23] J. J. Liu and D. M. Liu, Astrophys. Space Sci. 363(5), 115 (2018) doi: 10.1007/s10509-018-3331-0
    [24] J. J. Liu and D. M. Liu, New Astron. 69, 69 (2019a) doi: 10.1016/j.newast.2018.12.003
    [25] J. J. Liu, D. M. Liu, and L. H. Hao, ChPhC 43, 064107 (2019b) doi: https://iopscience.iop.org/article/10.1088/1674-1137/43/6/064107
    [26] J. J. Liu and D. M. Liu, Astron. Nachr. 341, 291 (2020) doi: https://onlinelibrary.wiley.com/doi/full/10.1002/asna.202013537
    [27] Q. H., Peng, J. J. Liu, and C. K. Chou, MPLA 35, 2050030 (2020) doi: 10.1142/S0217732320500303
    [28] Q. H. Peng, J. J. Liu, and Z. Q. Ma, New Astron. 57, 59 (2017) doi: 10.1016/j.newast.2017.06.011
    [29] W. Bernreuther and N. S. Craigie, Phys. Rev. Lett. 55, 2555 (1985) doi: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.55.2555
    [30] Q. H. Peng, Z. Y. Lie, and D. Y. Wang, Sci. Sin. Phys. Mech. As. 28, 970 (1985) doi: https://ui.adsabs.harvard.edu/abs/1985SSSMP..28..970P/abstract
    [31] E. Kolb and M. S. Turner., Astrophys. J. 286, 702 (1984) doi: 10.1086/162645
    [32] K. Freese, Asstrophys. J. 286, 216 (1984) doi: 10.1086/162589
    [33] Z. Q. Ma and J. F. Tang, Phys. Lett. B 153, 59 (1983) doi: https://linkinghub.elsevier.com/retrieve/pii/0370269383908158
    [34] T. Kallinger, W. W. Weiss, C. Barban et al., A&A 509, A77 (2010) doi: https://www.aanda.org/articles/aa/full_html/2010/01/aa11437-08/aa11437-08.html
    [35] D. Stello, and H. Bruntt, H. Preston et al., Astrophys. J. 674, L53 (2008) doi: 10.1086/528936
    [36] D. Huber, T. R. Bedding, D. Stello et al., Astrophys. J. 743, 143 (2011) doi: 10.1088/0004-637X/743/2/143
    [37] S. Sharma, D. Stello, and J. Bland-Hawthorn et al., Astrophys. J. 822, 15 (2016) doi: 10.3847/0004-637X/822/1/15
    [38] T. M. Brown, D. W. Latham, M. E. Everett et al., Astron. J. 142, 112 (2011) doi: 10.1088/0004-6256/142/4/112
    [39] E. Parker, Asstrophys. J. 160, 383 (1970) doi: 10.1086/150442
    [40] K. Freese and E. Krasteva, Phys. Rev. D 59, 063007 (1999) doi: https://journals.aps.org/prd/abstract/10.1103/PhysRevD.59.063007
    [41] S. K. Grunblatt, J. C. Zinn, A. M. Price-Whelan et al., Astrophys. J. 916, 88 (2021) doi: 10.3847/1538-4357/ac0532
    [42] V. Silva Aguirre, G. R. Davies, and S. Basu, MNRAS 452, 2127 (2015) doi: 10.1093/mnras/stv1388
    [43] T. D. Morton, Isochrones: Stellar Model Grid Package, v2.0, (Astrophysics Source Code Library, ascl: 1503.010, 2015)
    [44] D. Huber, J. Zinn, M. Bojsen-Hansen et al., Astrophys. J. 844, 102 (2017) doi: 10.3847/1538-4357/aa75ca
    [45] L. da Silva, L. Girardi, L. Pasquini et al., A&A 458, 609 (2006) doi: https://www.aanda.org/htbin/resolve?bibcode=2006A%26A...458..609DFUL
    [46] T. S. Rodrigues, D. Bossini, and A. Miglio, Mon. Not. R. Astron. Soc. 467, 1433 (2017) doi: https://academic.oup.com/mnras/article/467/2/1433/2917991
    [47] E. P. Bellinger, Mon. Not. R. Astron. Soc. 492, L50 (2020) doi: 10.1093/mnrasl/slz178
    [48] K. Belkacem, M. J. Goupil, M. A. Dupret et al., A&A 530, A142 (2011) doi: 10.1051/0004-6361/201116490
    [49] H. Kjeldsen and T. R. Bedding, A&A 293, 87 (1995)
    [50] D. Huber, D. Stello, T. R. Bedding et al., Comm. Asteroseismology 160, 74 (2009) doi: https://articles.adsabs.harvard.edu/pdf/2009CoAst.160...74H
    [51] K. Freese, M. S.Turner, and D. N. Schramm, Phys. Rev. Lett. 51, 1625 (1983) doi: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.51.1625
    [52] M. Schwarzschild, Structure and Evolution of the Star., (Princeton: Princeton university Press, 1958)
    [53] R. Bjork, Stephan, and B. Chaboyer, Astrophys. J. 641, 1102 (2006) doi: 10.1086/500505
    [54] D. A. Dicus, D. N. Page, and V. L. Teplitz, Phys. Rev. D 26, 1306 (1982) doi: https://journals.aps.org/prd/abstract/10.1103/PhysRevD.26.1306
    [55] L. G. Althaus, A. H. Córsico, J. Isern et al., Astron. Astrophys. Rev. 18, 471 (2010) doi: https://link.springer.com/article/10.1007/s00159-010-0033-1
    [56] N. Itoh, H. Hayashi, A. Nishikawa et al., Astrophys. J. Suppl. 102, 411 (1996) doi: 10.1086/192264
    [57] M. Izawa, Prog. Theor. 75, 556 (1986) doi: https://academic.oup.com/ptp/article/75/3/556/1915730

  • Access

    加载中

Figures(2) / Tables(11)

Get Citation
Jing-Jing Liu, Dong-Mei Liu and Liang-Huan Hao. New insights on the limit of the magnetic monopole flux and the heating source in white dwarfs[J]. Chinese Physics C. doi: 10.1088/1674-1137/acdc8b
Jing-Jing Liu, Dong-Mei Liu and Liang-Huan Hao. New insights on the limit of the magnetic monopole flux and the heating source in white dwarfs[J]. Chinese Physics C.  doi: 10.1088/1674-1137/acdc8b shu
Milestone
Received: 2023-04-09
Article Metric

Article Views(1269)
PDF Downloads(31)
Cited by(0)
Policy on re-use
To reuse of subscription content published by CPC, the users need to request permission from CPC, unless the content was published under an Open Access license which automatically permits that type of reuse.
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Email This Article

Title:
Email:

New insights into the limit of the magnetic monopole flux and the heating source in white dwarfs

  • College of Science, Hainan Tropical Ocean University, Sanya 572022, China
  • Received Date: 2023-04-09
  • Available Online: 2023-08-15

Abstract: Based on the magnetic monopole (MM) catalytic nuclear decay (Rubakov-Callan (RC) effect), we propose five new models to discuss the limit of the MM flux and the heating energy resources of white dwarfs (WDs) based on observations of 13 red giant branch (RGB) stars. We find that the number of MMs captured can reach a maximum value of 9.1223×1024 when m=1017 GeV, nB=5.99×1031 cm−3, ϕ=7.59×10−26cm−2s−1sr−1. The good agreement of our calculated luminosities for WDs with observation provides support for our model based on the RC effect by MMs. We obtain a new limit of the MM flux of ξ=ϕ⟨σmvT⟩−28≤9.0935×10−13cm−2s−1sr−1, and ξ≤4.9950×10−13cm−2s−1sr−1 at nB=5.99×1031cm−3 when m=1015GeV, β=9.4868×10−3, and m=1017GeV, β=10−3, respectively. Our results show that the RC effect could cause heating that prevents white dwarfs from cooling down into a stellar graveyard. Our results will also provide a new idea for further research on the upper limit of MM flow (note: nB,σm,m,ϕ,ξ are the baryon number density, reaction cross section, mass, MM flux, and the new limit of the MM flux, respectively, and β=vT/c is the ratio of the speed of MMs to that of light).

    HTML

    I.   INTRODUCTION

    • White dwarfs (hereafter WDs) are usually made up of C+O. However, it is also possible for their cores to be hot enough to burn carbon but not hot enough to burn Ne, forming a WD with a core of O+Ne+Mg. At the later stage of WDs, the star ejects large quantities of matter. After great mass loss, if the remaining core mass is less than 1.44 solar masses, the star may evolve into a WD.

      WDs form at very high temperatures. Because they have no source of energy, they will therefore gradually give off heat and cool down, whereupon its radiation will decrease over time from its initial high color temperature to red. This surface temperature is defined in astronomy as the effective temperature Teff as per Stefan's law, so that

      Lrad=4πR2σT4eff,

      		(1)

      where R the radius of the star, Lrad is the radiation luminosity, and σ=5.6704×10−5ergs−1cm−2K−4 is the radiation constant from Stefan's law. Teff is a measure of the energy flux at the surface and not a real temperature, but it nevertheless constitutes a useful measure of the atmospheric temperature of the star.

      As is well known, the effective temperature of WDs is mostly in the range 5500–40000 K, while a few are outside this range, and the internal temperature of WDs is on the order of ∼106−107K, with a total thermal energy less than 1047ergs. Mestel. [1] discussed the energy sources of WDs. Avakian. [2] also studied the configurations of hot WDs with nuclear sources of energy. Bildsten & Hall. [3] discussed the sources of white dwarfs, suggesting that there are small amounts of 22Ne in some WDs that may constitute an extra source of heat in carbon-oxygen WDs. Single-particle 22Ne sedimentation may be considered a possible heat source [4, 5]. However, some work suggests that 22Ne must separate into clusters, enhancing diffusion, in order for sedimentation to provide heating on the observed timescale. Recently, the sources of ultra-high-energy photons for WD pulsars have been discussed by Lobato et al. [6]. Cheng et al. [7] discuss the cooling anomaly of high-mass WDs, pointing out that 22Ne settling in C/O-core WDs could account for this extra cooling delay. Caplan et al. [8] studied this topic using molecular dynamics methods and phase diagrams, from which they ruled out the isotope 22Ne as a possible cause of the extra heating. Therefore, the problem of additional heat sources for WDs remains a challenging topic.

      In this paper, we selected 13 red giant branch (RGB) stars to present a new model of the number of magnetic monopoles (hereafter MMs) captured, and seek to solve the energy source problem for WDs based on MM catalytic nuclear decay (the Rubakov-Callan (RC) effect) [9, 10]. MMs are hypothetical magnetic particles with a single north or south magnetic pole, which have been proposed in string theory. Research on MMs has long been a hot topic among physicists and astronomers. Some papers have discussed the issues of MMs, (e.g., Callan [9], Detrixhe et al. [11], Frank et al. [12], Fujii & Pierre [13], Kain [14], Rajantie [15]). Recently, Mavromatos & Mitsou. [16] discussed the developments in both theory and experimental searches for MMs in past, current, and future colliders and in the cosmos. We also are interested in the problem of MMs and other related issues ((e.g., Liu. [17, 18], Liu & Gu [19], Liu et al. [20], Liu [21−26], Peng et al. [27, 28]).

      The arrangement of this paper is as follows. In the next section, we discuss the number of possible magnetic monopoles in space and the luminosity due to the RC effect by MMs. In Section III, we describe our models and the luminosity function due to magnetic monopole catalytic nuclear decay. In Section IV, some results and discussions are presented. Finally, our conclusions are summarized in Section V.

    II.   THE NUMBER OF MMs CAPTURED IN SPACE AND THE LUMINOSITY DUE TO THE RC EFFECT BY MMs

    • According to some research, the interaction of MMs with neutral hydrogen atoms is very weak. Therefore, during the process of formation of celestial bodies, very few MMs are captured in the collapse of a neutral hydrogen cloud and collect in the core of a star or planet. MMs typically may be contained within stars and planets, and they are mostly captured from space during their lifetime after formation. One type of interaction that MMs undergo in stars and planets might be the RC effect, through which MMs may catalyze nucleon decay, as expressed by p+M→e+π0+M+debris (x%) and p+M→e+μ±+M+debris (y%). The ratio of the cross sections of the above reactions x/y may be a few percent, which is on the order of ∼10−4 [10]. Bernreuther and Craigie [29] discussed the cross section of monopole-induced proton decay in SU(5) for the above reactions, showing that the ratio of cross sections of the above reactions x/y may be (2.5∼2.8)×10−4/(0.1∼0.3).

      The number (flow) of MMs captured in space at the surface of stars and planets (including earth) is estimated as follows [30]:

      Nm=4π2R2ηϕt[1+(vescvm)2],

      		(2)

      where ϕ is the flux of MMs intercepted in space and vesc=(2GM/R)1/2, t are the escape velocity from the star and cooling age of the star, respectively. vm represents the velocity of MMs in space, For a planet, vescvm≪1, and vescvm≈1 for ordinary stars, but vescvm≫1 for compact objects such as WDs, neutron stars, galactic nuclei, and quasars. Therefore, the ratio between the number of MMs captured and the number of nuclei in the MMs accumulation area is [30]

      NmNB=5.10×10−34ηR2∗(t9M∗)(ϕϕ0)[1+4.256v−2−3(M∗R∗)],

      		(3)

      where R∗=R/R⊙, M∗=M/M⊙, v−3=vm/10−3c, t9=t/109Yr, and c is the speed of light. ϕ0≈10−12cm−2s−1Sr−1 [31]. η is the probability of the capture of MMs by stars, which depends on the ratio of the penetration distance lpd of an MM in a star to the star's radius. In general, we have lpd≈1.2×1030v−3n−1eT1/2e for plasma [30]. For example, with ne∼1022cm−3,Te∼106K for the sun, we have lpd∼1011,η∼0.7; however, for WDs and neutron stars, ne≥1030,ne≥1035cm−3, respectively, and η∼1. For quasars and active galactic nuclei, M∗∼108,Te∼105K, η∼1.

      The velocity of MMs is also determined as a function of monopole mass by βT as follows [32].

      vm=cβT={3×10−3c(1016GeVm)0.5(m<1017GeV)10−3c(otherwise).

      		(4)

      According to Eq. (3), for WDs we have η∼1; thus, the total number of magnetic monopoles trapped in space after the formation of stars (or planets) is estimated to be

      Nm=7.18×1011nBR5∗ϕ(t9M∗)[1+4.256v−2−3(M∗R∗)].

      		(5)

      For astronomy, the most important property of a magnetic monopole is that it can trigger the RC effect, as independently proposed by Rubakov and Callen [9, 10]. The reaction cross section is about σm≈10−25∼10−26cm2, almost reaching the Thomson cross section (6.665×10−25cm2). The luminosity of various types of celestial bodiesdue to the RC effect (i.e., RC luminosity) can be estimated as follows. In the core area, where the magnetic monopole is concentrated, the nuclear decay reaction is catalyzed by the magnetic monopoles and the total luminosity produced is [30]

      Lm≈4π3r3cnmnB⟨σmvT⟩mBc2=NmnB⟨σmvT⟩mBc2,

      		(6)

      where rc, and nm,nB are the radius of the stellar central region and the number densities of MMs and nucleons, respectively.

      In Eq. (6), vT is the thermal movement speed of the nucleus relative to the magnetic monopole. We will ignore the thermal velocity of the magnetic monopole due to its great mass. We thus only consider the contributions from the thermal velocity of the nucleus. According to 1/2mv2T=3/2kT, we have vT=√3kT/mB≈1.5745×107T1/26cm/s, where T is the temperature, T6=T/106K, k=1.38×10−16erg/s is the Boltzmann constant, and mB≈1.67×10−24g is the nucleon mass. As the central temperature of WDs is about ∼106K, we have vT∼10−3c.

      As a general rule, the reaction cross section of the RC effect σm is 10−26∼10−24cm2. Ma & Tang [33] gave a value σm≈4.28676×10−24cm2 for the cross section of the proton with different channels using SU(5) grand unification theory. In the RC process, MMs induced nucleon decay, followed by nucleon decay into π0 mesons, μ± leptons, and positrons e+, and μ± and π0 again decay into photons and electron-positron pairs e±. The positrons then undergo annihilation with electrons to form photons. The net effect is that the rest mass energy of nucleons (mBc2) is entirely converted to radiant energy with 100% efficiency (1mBc2≈1GeV≈1.6×10−3ergs).

    III.   THE MAGNETIC MONOPOLE MODEL AND RC LUMINOSITY INSIDE WHITE DWARFS

      A.   The mass and radius for the red giant branch (RGB) phase

    • It is well known that WDs evolve from red giants with mass less than 8M⊙. The MM content of the trapping accumulation set is mainly from the red giant branch (RGB) phase. The M∗ and R∗ for RGB stars is given by [34, 35]

      Mr∗=MRGM⊙=(νmaxνmax,⊙)3(ΔνfΔνΔν⊙)−4(TeffTeff,⊙)1.5,

      		(7)

      Rr∗=RRGR⊙=(νmaxνmax,⊙)(ΔνfΔνΔν⊙)−2(TeffTeff,⊙)0.5,

      		(8)

      where νmax,⊙=3090μHz, Δν⊙=135.1μHz, Teff,⊙=5777K, and Δν≃α(νmax/μHz)β (here α=0.268μHz, β=0.758)[36]. fΔν is a typical asteroseismic correction factor, which is between 0.98 and 1.02 [37]. The predicted νmax values are calculated using the scaling relation [38]

      νmaxνmax⊙=(gg⊙)(TeffTeff,⊙)−0.5.

      		(9)

      where g is the surface gravity of the star and g⊙=27487cm/s2.

      • B.   The magnetic monopole RC effect model in white dwarfs

    • According to Eqs. (5)–(9), the number of magnetic monopoles captured from space and the total luminosity due to the RC effect by the MMs are given by

      Nm=7.18×1011nB(Rr∗)5ϕ(t9Mr∗)[1+4.256v−2−3(Mr∗Rr∗)].

      		(10)

      Lm≈4π3r3cnmnB⟨σmvT⟩mBc2=NmnB⟨σmvT⟩mBc2,=1.15×109n2B(Rr∗)5ϕt9(Mr∗)−1×⟨σmvT⟩[1+4.256v−2−3(Mr∗Rr∗)].

      		(11)

      Defining ξ=ϕ⟨σmvT⟩−28=ϕ⟨σmvT⟩/10−28, Eq. (11) may be rewritten

      Lm≈4π3r3cnmnB⟨σmvT⟩mBc2=NmnB⟨σmvT⟩mBc2,=1.15×1037n2B(Rr∗)5t9(Mr∗)−1ξ[1+4.256v−2−3(Mr∗Rr∗)].

      		(12)

      From Eqs. (1) and (12), and Lm=Lrad, we obtain

      ξ=5.29×10−15σMr∗t−19n−2B(Rr∗)−3T4eff[1+4.256v−2−3(Mr∗Rr∗)]−1.

      		(13)
    IV.   RESULTS AND DISCUSSIONS

    • The study of MMs has held considerable interest since MMs were found to be a generic feature of grand unified gauge theories in the physical fields. The theoretical predictions of monopole abundance are problematic in the standard cosmology, as far too many monopoles would have survived annihilation for the universe to have reached its present state. For example, the galactic field that yields the Parker bound is ϕ(σv)−28≤10−16cm−2s−1sr−1 [39]. Due to MM RC decay, another limit of the flux may be ϕ(σv)−28≤10−21cm−2s−1sr−1[31]. A better-understood limit in WDs may be ϕ(σv)−28≤10−18cm−2s−1sr−1 [32]. Then the bound has been stated as ϕ(σv)−28≤10−28cm−2s−1sr−1 by Freese and Krasteva [40]. In this paper, we study MMs and their numbers in space and discuss our MM model and the luminosity due to RC effect. We select the following typical parameters: m=1015,1017GeV, σm=10−24cm2, ϕ=5.59×10−28,10−26,7.59×10−26cm−2s−1sr−1, and nB=5.99×1031,1.89×1032cm−3, t9=1,10.

      Everyone knows that WDs evolve from red giants with mass less than 8 M⊙, we focus on 13 typical RGB stars listed in Tables 1 and 2 [41] and discuss their mass, radius, and cooling age. Depending on an asteroseismic correction factor, we propose five models (I−V) to examine the problem of the energy source of WDs.

      starTIC IDGaia IDGania magDistance (pc)Galactic substructureSpectral source
      1TIC20897763Gaia 23656494710338280969.41484457.879 ± 9.435Gaia Enceladus SausageAPOGEE
      2TIC341816936Gaia 142177604633572300811.6231547.85 ± 45.63Gaia Enceladus SausageAPOGEE
      3TIC393961551Gaia 15063876279179368969.57166500.533 ± 6.312Gaia Enceladus SausageAPOGEE
      4TIC453888381Gaia 523025673034745715210.9714788.614 ± 15.999HaloGALAH
      5TIC279510617Gaia 548055045064301721610.7551933.263 ± 22.520HaloGALAH
      6TIC300938910Gaia 527067501829784422410.5629607.156 ± 7.7075HaloGALAH
      7TIC198204598Gaia 162989868534727385610.9455952.885 ± 37.512Halo···
      8TIC1008989Gaia 37896392809526103049.72882370.56 ± 5.85Gaia Enceladus Sausage···
      9TIC91556382Gaia 506500965033314739210.0855870.289 ± 34.565Halo···
      10TIC159509702Gaia 170919509028171827212.15421595.67 ± 55.605Halo···
      11TIC25079002Gaia 46693160657002229769.91465716.804 ± 15.5995DiskAPOGEE
      12TIC177242602Gaia 526229539536721228810.1451532.831 ± 14.294DiskAPOGEE
      13TIC9113677Gaia 324548565060765158410.1791491.438 ± 10.040Thick Disk···

      Table 1.  Information of the 13 red giant stars selected [41].

      starνmaxΔν /μHzTeff /K[Fe/H][α/Fe]
      161.31383 ± 1.217686.81739 ± 0.249904988 ± 127−1.274 ± 0.0190.219 ± 0.021
      236.34 ± 0.764.2969 ± 0.07155068 ± 100−1.873 ± 0.1070.248 ± 0.023
      361.34 ± 1.756.68 ± 0.415121 ± 105−1.0751 ± 0.01230.156 ± 0.014
      450.37 ± 1.595.953 ± 0.0494741 ± 100−0.728 ± 0.070.32 ± 0.021
      528.57 ± 0.163.566 ± 0.0154450 ± 100−0.49 ± 0.050.281 ± 0.017
      6106.30 ± 0.929.464 ± 0.1314908 ± 100−0.792 ± 0.050.2566 ± 0.0165
      745.86 ± 0.315.132 ± 0.0324979 ± 100······
      8104.33059 ± 1.466189.80317 ± 0.153364893 ± 100······
      931.38665 ± 1.030974.189 ± 0.1865192 ± 100······
      1045.37 ± 0.535.090 ± 0.0274724 ± 100······
      1145.238 ± 0.624.967 ± 0.1214797 ± 830.1636 ± 0.006−0.010 ± 0.006
      1254.66 ± 0.335.663 ± 0.0314603 ± 100−0.1176 ± 0.0060.125 ± 0.007
      1387.22437 ± 0.870428.365 ± 0.1984764 ± 100······

      Table 2.  Information of the 13 RGB stars [41].

      In this paper, we select the asteroseismic correction factors f△ν= 0.98, 0.99, 1.00, 1.01, 1.02 for our typical models under study, which correspond to models (I − V). Based on Eqs. (8) and (9), we can calculate the mass and the radii of RGB stars Mr∗ (I − V) and Rr∗ (I − V) given in Table 3. We determine the stellar (RGB) ages using the packages BASTA [42], isochrones [43], isoclassify [44], PARAM [45, 46], and scaling-giants [47]. These ages correspond to t9 (I ∼ V) in Table 4.

      starM∗(I)M∗(II)M∗(III)M∗(IV)M∗(V)R∗(I)R∗(II)R∗(III)R∗(IV)R∗(V)
      10.89160.92860.96671.00591.04646.95417.09677.24087.38637.5333
      21.20471.25461.30611.35911.413810.45810.67210.88911.10811.329
      31.00751.04931.09231.13671.18247.34217.49277.64497.79857.9537
      40.78790.82060.85420.88890.92467.30457.45437.60567.75857.9129
      51.01541.05751.10091.14561.191611.18611.41611.64711.88212.118
      61.22111.27171.32391.37761.43306.20576.33306.46166.59146.7226
      71.15871.20671.25621.30721.35989.17039.35849.54849.74049.9342
      80.98961.04041.08301.12701.17235.66995.78625.90376.02246.1422
      90.89100.92800.96601.00521.04579.61939.816610.01610.21710.421
      100.92210.96030.99971.04031.08218.54488.72018.89729.07609.2566
      111.19861.24831.29951.35231.40669.47889.67329.869610.06810.268
      121.17621.22491.27521.32701.38038.63088.80788.98679.16739.3497
      131.05871.10261.14781.19451.24256.42506.55686.68996.82446.9602

      Table 3.  The mass and radii of the 13 RGB stars selected [41]. The asteroseismic correction factor f△ν= 0.98, 0.99, 1.00, 1.01, 1.02 defines models I − V, respectively.

      starscaling-giants Age (Gyr)isochrones Age (Gyr)isoclassify Age (Gyr)PARAM Age (Gyr)BASTA Age (Gyr)
      t9(I)t9(II)t9(III)t9(IV)t9(V)
      15.8 ± 3.08.77 ± 2.95.68+0.79−0.839.29+2.82−3.239.0+2.7−2.5
      22.9 ± 1.89.16 ± 2.757.44+1.36−1.346.52+4.46−3.807.9+2.8−1.9
      34.7 ± 6.05.93 ± 3.055.63+0.84−0.695.72+4.61−3.147.5+3.3−2.2
      410.5 ± 3.59.72 ± 2.5012.78+0.78−1.5911.66+1.50−2.4214.8±2.8
      56.4 ±1.17.99 ± 3.3710.23+0.95−2.347.68+4.55−4.457.6+0.9−0.8
      62.7 ± 0.83.693 ± 0.7023.34+0.40−0.332.11+0.61−0.492.9+0.6−0.4
      73.3 ± 0.88.749 ± 2.1611.54+0.66−2.831.98+7.13−0.775.8+1.6−1.2
      85.8 ± 2.06.843 ± 3.336.84+1.50−0.907.27+2.95−2.0910.6+3.5−2.9
      97.1 ± 5.06.841 ± 3.596.43+1.89−0.965.27+2.84−1.778.9+3.8−2.9
      104.8 ± 1.29.344 ± 2.3319.98+1.47−1.187.92+3.47−2.4910.0+3.2−2.5
      112.1 ± 3.27.07 ± 3.651.92+0.17−0.111.84+0.18−0.143.6 ^{+0.3}_{-0.4}
      124.4 \pm 2.77.49 \pm 3.456.54 ^{+0.44}_{-0.64} 7.60 ^{+4.02}_{-3.74} 6.4 ^{+0.6}_{-1.0}
      135.1 \pm 2.07.07 \pm 3.656.77 ^{+1.9}_{-1.35} 8.24 ^{+3.24}_{-2.87} 9.7 ^{+3.3}_{-2.3}

      Table 4.  The cooling age of the 13 RGB stars selected [41]. The five cooling ages correspond to models I - V. The discussions of the cooling ages are based on the packages BASTA [42], isochrones [43], isoclassify [44], PARAM [45, 46], and scaling-giants [47].

      Based on Eqs. (7)–(9), there are two global seismic parameters, the frequency of maximum power, \nu_{\rm{max}} , and the mean large frequency separation, \bigtriangleup \nu , to describe the oscillations of solar-like stars and radius. The surface gravity and temperature strongly determine the value of the frequency of maximum power, which is given by \nu_{\rm{max}}\propto gT^{-0.5}_{\rm{eff}}\propto MR^{-2}T^{-0.5}_{\rm{eff}} [48]. On the other hand, the travel time of sound from the center to the surface of a star will directly influence \bigtriangleup \nu , which is sensitive to the mean stellar density and is given by \bigtriangleup \nu\propto \rho^{0.5}\propto M^{0.5}R^{-1.5} [49]. In order to reduce systematic errors, by considering the effect on mass, [Fe/H] and T_{\rm{eff}} interpolation over a grid of models, a modification strategy was adopted by Sharma et al. [37]. Therefore, the asteroseismic correction factor in this paper is selected as f_{\bigtriangleup \nu}= 0.98, 0.99, 1.00, 1.02, we propose five model (I-V). Detailed information on the mass and radius of RGB stars is shown in Table 3.

      Based on Eqs. (10) and (12), the age of RGB stars is a key parameter for estimating the number of MMs captured and the luminosity of WDs due to the RC effect by MMs. By combining masses inferred from the asteroseismic parameters with stellar atmospheric parameters and using stellar isochrones, the cooling ages of RGB stars were determined as shown in Table 4 by t_9 . Using the packages BASTA [42], isochrones [43], isoclassify [44], PARAM [45, 46], and scaling-giants [47], estimates are obtained of the cooling ages of RGB stars, which are the stellar ages since zero-age main sequence stars are considered.

      The scaling-giants package accepts asteroseismic parameters, metallicity, and temperature as inputs for model I. In the isochrones, isoclassify, PARAM, and BASTA packages, we take asteroseismic, photometric, and spectroscopic parameters as inputs for models II−V. From the SYD pipeline [50], along with effective temperatures determined through the direct method of isoclassification and metallicity determined by either the APOGEE or GALAH surveys, and using measured seismic \nu_{\rm{max}} and \bigtriangleup\nu values, we can determine the ages with scaling-giants for model I. The SYD pipeline is an automated pipeline to estimate global oscillation parameters, such as the frequency of maximum power ( \nu_{\rm{max}} ) and the large frequency spacing ( \triangle\nu ), for a large number of time series. By using 2MASS K magnitudes, asteroseismic \bigtriangleup\nu , Gaia parallaxes, and temperatures and metallicities from spectroscopy, we can determine the ages with isochrones, isoclassify, and PARAM, and BASTA packages for models II−V, respectively. Some useful parameters are presented in Tables 1–3. Detailed discussions are given in Grunblatt et al. [41].

      Based on Eq. (10), we give a computational assessment for the number of MMs captured for WDs in Tables 5 and 6 for m=10^{15}, 10^{17} GeV, respectively. Our results in Table 5 show that the maximum number of MMs captured in models I \sim V are 4.8126\times10^{22} and 6.5345\times10^{24} when \phi=5.59\times10^{-28} \;\rm{cm^{-2}s^{-1}sr^{-1}} , and \phi=7.59\times 10^{-26} \rm{cm^{-2}s^{-1}sr^{-1}} , respectively. Table 6 also shows that the number of MMs captured has a maximum of 9.1223\times10^{24} (e.g., when m=10^{17}{\rm GeV}, n_{\rm B}=5.99\times 10^{-31} \rm{cm^{-3}}, \phi=7.59\times10^{-26} \rm{cm^{-2}s^{-1} sr^{-1}}, for model III). One can see that the number of MMs captured increases as the flux of MMs increases due to N_m\propto \phi according to Eq. (10). On the other hand, when the flux and mass of MMs are certain, there is no significant difference found in the number of MMs captured for same stars among the five models. However, for different models, there is a small difference. The reasons for these differences can be from the differences of the cooling ages for RGB stars due to differences in the stellar parameters selected, such as the asteroseismic correction factor and [\rm{Fe/H}] . From the above analysis, it may be seen that the number of MMs captured can reach the maximum value of 9.1223\times10^{24} when m=10^{17}{\rm GeV}, n_{\rm B}= 5.99\times10^{-31}\rm{cm^{-3}}, \phi= 7.59\times10^{-26} \rm{cm^{-2}s^{-1}sr^{-1}}.

      \phi=5.59\times10^{-28}\rm{cm^{-2}s^{-1}sr^{-1}} \phi=7.59\times10^{-26}\rm{cm^{-2}s^{-1}sr^{-1}}
      star N_m (I) N_m (II) N_m (III) N_m (IV) N_m (V) N_m (I) N_m (II) N_m (III) N_m (IV) N_m (V)
      12.5599e214.1144e212.8307e214.9152e215.0524e213.4758e235.5864e233.8435e236.6738e236.8601e23
      27.2823e212.4449e222.7645e221.9626e222.5231e229.8877e233.3197e243.7536e242.6648e243.4258e24
      32.4095e213.2314e213.2591e213.5154e214.8906e213.2716e234.3876e234.4251e234.7731e236.6404e23
      46.6993e216.5919e219.2068e218.9176e211.2010e229.0962e238.9503e231.2501e241.2108e241.6307e24
      52.6667e223.5386e224.8126e223.8356e224.0272e223.6207e244.8046e246.5345e245.2079e245.4680e24
      64.9402e207.1828e206.9014e204.6290e206.7508e206.7077e229.7527e229.3705e226.2851e229.1661e22
      74.4692e211.2594e221.7647e221.4790e229.9908e216.0681e231.7100e242.3961e242.0081e241.3565e24
      88.2517e201.0349e211.0989e211.2401e211.9185e211.1204e231.4051e231.4921e231.6837e232.6049e23
      91.5855e221.6237e221.6212e221.4106e222.5275e222.1527e242.2046e242.2012e241.9152e243.4318e24
      105.7326e211.1862e221.3458e221.1338e221.5189e227.7836e231.6105e241.8273e241.5395e242.0624e24
      113.2438e211.1608e223.3488e213.4071e217.0729e214.4044e231.5761e244.5469e234.6261e239.6034e23
      124.3369e217.8473e217.2788e218.9801e218.0238e215.8885e231.0655e249.8830e231.2193e241.0895e24
      131.2784e211.8839e211.9164e212.4764e213.0932e211.7359e232.5579e232.6020e233.3624e234.1999e23

      Table 5.  The number of MMs captured in the five models I-V (corresponding to typical asteroseismic correction factor values f_{\bigtriangleup\nu}=0.98, 0.99, 1.00, 1.01, 1.02 ) when m=10^{15}{\rm GeV}, n_{\rm B}=5.99\times10^{31}\rm{cm^{-3}}, \phi=5.59\times10^{-28}, 7.59\times10^{-26}\rm{cm^{-2}s^{-1}sr^{-1}}.

      \phi=5.59\times10^{-28}\rm{cm^{-2}s^{-1}sr^{-1}} \phi=7.59\times10^{-26}\rm{cm^{-2}s^{-1}sr^{-1}}
      star N_m (I) N_m (II) N_m (III) N_m (IV) N_m (V) N_m (I) N_m (II) N_m (III) N_m (IV) N_m (V)
      13.9330e216.3662e214.4113e217.7145e217.9867e215.3401e238.6439e235.9895e231.0475e241.0844e24
      21.0794e223.6479e224.1522e222.9675e223.8405e221.4656e244.9531e245.6377e244.0292e245.2146e24
      33.7921e215.1235e215.2058e215.6570e217.9291e215.1488e236.9565e237.0683e237.6810e231.0766e24
      49.7253e219.6300e211.3536e221.3195e221.7884e221.3205e241.3075e241.8379e241.7915e242.4282e24
      53.6811e224.9121e226.7186e225.3850e225.6862e224.9981e246.6696e249.1223e247.3117e247.7206e24
      68.9937e201.3196e211.2795e218.6609e201.2747e211.2212e231.7918e231.7373e231.1760e231.7307e23
      76.8316e211.9388e222.7358e222.3091e221.5709e229.2759e232.6324e243.7146e243.1352e242.1330e24
      81.4320e211.8114e211.9401e212.2082e213.4458e211.9443e232.4595e232.6342e232.9982e234.6787e23
      92.2008e222.2668e222.2762e221.9920e223.5898e222.9883e243.0778e243.0906e242.7046e244.8742e24
      108.3230e211.7331e221.9789e221.6778e222.2622e221.1301e242.3532e242.6868e242.2781e243.0715e24
      114.9599e211.7875e225.1930e215.3209e211.1124e226.7345e232.4270e247.0510e237.2247e231.5105e24
      126.8084e211.2410e221.1597e221.4414e221.2975e229.2444e231.6851e241.5746e241.9571e241.7617e24
      132.1582e213.2067e213.2891e214.2856e215.3977e212.9304e234.3540e234.4659e235.8190e237.3289e23

      Table 6.  The number of MMs captured in the five models I-V (corresponding to typical asteroseismic correction factor values f_{\Delta\nu}=0.98, 0.99, 1.00, 1.01, 1.02 ) when m=10^{17}{\rm GeV}, n_{\rm B}=5.99\times10^{31}\rm{cm^{-3}}, \phi=5.59\times10^{-28}, 7.59\times10^{-26}\rm{cm^{-2}s^{-1}sr^{-1}}.

      Figures 1 and 2 display the luminosities as a function of t_9 of WDs of the five models under different astronomical conditions. From the two figures, it can be seen that the calculated luminosities agree well with observations. One can also conclude the same from Table 7. For example, the ranges of our calculated luminosities are 1.3336\times10^{34}\sim 7.1985\times10^{36}\rm{erg \; s^{-1}} and 1.8224\times10^{34}\sim 1.0871\times10^{36}\rm{erg\; s^{-1}} for models I and V, respectively, in Table 7.

      Figure 1.  (color online) The luminosities as a function of t_9 for the 13 WDs [41] for the five models I -V when n_{\rm B}=5.99\times10^{31}\rm{cm^{-3}}, m=10^{15} GeV, \sigma_m=10^{-24}\rm{cm^2}, \phi=10^{-26}\rm{cm^{-2}s^{-1}sr^{-1}} at the temperature T_6=1 .

      Figure 2.  (color online) The luminosities as a function of t_9 of the 13 WDs [41] for models I - V when n_{\rm B}=5.99\times10^{31}\rm{cm^{-3}}, m=10^{15} GeV, \sigma_m=10^{-24}\rm{cm^2}, \phi=10^{-26}\rm{cm^{-2}s^{-1}sr^{-1}} at the temperature T_6=10 .

      starour resultsobservations
      L_{m }(I)L_{m }(II)L_{m }(III)L_{m }(IV)L_{m }(V) L_{\rm{rad}} (I) L_{\rm{rad}} (II) L_{\rm{rad}} (III) L_{\rm{rad}} (IV)L_{m }(V)
      16.9103e341.1107e357.6414e341.3269e351.3639e351.0318e351.0746e351.1187e351.1641e351.2109e35
      21.9658e356.6000e357.4626e355.2980e356.8110e352.4869e352.5900e352.6962e352.8057e352.9185e35
      36.5043e348.7231e348.7978e349.4896e341.3202e351.2778e351.3308e351.3854e351.4417e351.4996e35
      41.8085e351.7795e352.4853e352.4073e353.2420e359.2913e349.6763e341.0073e351.0482e351.0904e35
      57.1985e359.5522e351.2992e361.0354e361.0871e361.6913e351.7614e351.8336e351.9081e351.9848e35
      61.3336e341.9390e341.8630e341.2496e341.8224e347.7022e348.0214e348.3505e348.6895e349.0388e34
      71.2064e353.3998e354.7637e353.9925e352.6970e351.7814e351.8552e351.9313e352.0097e352.0905e35
      82.2275e342.7936e342.9665e343.3475e345.1790e346.3515e346.6147e346.8861e347.1657e347.4537e34
      94.2799e354.3831e354.3763e353.8078e356.8228e352.3176e352.4137e352.5127e352.6147e352.7198e35
      101.5475e353.2020e353.6329e353.0607e354.1003e358.3978e348.7458e349.1046e349.4742e349.8551e34
      118.7565e343.1336e359.0399e349.1974e341.9093e351.6398e351.7078e351.7779e351.8501e351.9244e35
      121.1707e352.1183e351.9649e352.4242e352.1660e351.1526e351.2004e351.2496e351.3004e351.3526e35
      133.4511e345.0855e345.1732e346.6849e348.3500e347.3599e347.6649e347.9794e348.3034e348.6371e34

      Table 7.  Comparisons of the calculated luminosities (L_{m}) due to the RC effect by MMs with observations ( L_{\rm{rad}} ) for the five models I - V at n_{\rm B}=5.99\times10^{31}{\rm cm^{-3}}, m=10^{15}GeV, \sigma_m=10^{-24}{\rm cm^2}, \phi=10^{-26}{\rm cm^{-2}s^{-1}sr^{-1}}, T_6=1 ( \xi=1.5745\times10^{-15}\rm{cm^{-2}s^{-1}sr^{-1}} ).

      In order to facilitate comparison of our results with the observed data, we define the scale factor k_i as the ratio of our results due to the RC effect to the observed luminosity. We note that the largest differences between our results and the observed values are one order of magnitude. For example, based on Table 7 and Table 9 for star 5, the maximal scale factors are k_1=4.2562 , k_2=5.4231 , k_3=7.0851 , k_4=5,4264 , and k_5=5.4772 for models I−V, respectively. However, in Table 8 and Table 9, the maximal scale factors for star 5 are k_1=13.459 , k_2=17.149 , k_3=22.405 , k_4=17.160 and k_5=17.321 for models I−V, respectively.

      starTable 7Table 8
      k_1 k_2 k_3 k_4 k_5 k_1 k_2 k_3 k_4 k_5
      10.669731.03360.683091.13981.12642.11793.26852.16013.60453.5619
      20.790482.54832.76781.88832.33382.49978.05858.75275.97147.3801
      30.50900.65550.63500.65820.88041.60962.07282.00822.08152.7840
      41.94641.83902.46732.29652.97336.15515.81537.80227.26239.4024
      54.25625.42317.08515.42645.477213.45917.14922.40517.16017.321
      60.17310.24170.22310.14380.20160.547530.76440.70550.45470.6376
      70.67731.83262.46661.98661.29012.14165.79517.80006.28214.0797
      80.35070.422330.43080.46720.69481.10901.33551.36231.47732.1972
      91.84671.81591.74171.45632.50855.83965.74255.50764.60527.9327
      101.84273.66123.99023.23064.16065.827311.57812.61810.21613.157
      110.53401.83480.50850.49710.99221.68865.80231.60791.57213.1374
      121.01571.76471.57241.86421.60133.21205.58064.97235.89525.0638
      130.46890.66350.64830.80510.96681.48282.09812.05022.54593.0572

      Table 9.  The comparisons of the scale factor k_i ( i=(1-5) indexes models I−V), which is the ratio of our results (L_{m}) due to RC effect by MMs to the observed luminosities ( L_{\rm{rad}} ) for the five models I−V from Tables 7 and 8.

      starour resultsobservations
      L_{m }(I)L_{m }(II)L_{m }(III)L_{m }(IV)L_{m }(V) L_{\rm{rad}} (I) L_{\rm{rad}} (II) L_{\rm{rad}} (III) L_{\rm{rad}} (IV)L_{m }(V)
      12.1852e353.5122e352.4164e354.1959e354.3130e351.0318e351.0746e351.1187e351.1641e351.2109e35
      26.2165e352.0871e362.3599e361.6754e362.1538e362.4869e352.5900e352.6962e352.8057e352.9185e35
      32.0568e352.7585e352.7821e353.0009e354.1749e351.2778e351.3308e351.3854e351.4417e351.4996e35
      45.7189e355.6271e357.8593e357.6125e351.0252e369.2913e349.6763e341.0073e351.0482e351.0904e35
      52.2764e363.0207e364.1083e363.2743e363.4378e361.6913e351.7614e351.8336e351.9081e351.9848e35
      64.2172e346.1316e345.8913e343.9515e345.7628e347.7022e348.0214e348.3505e348.6895e349.0388e34
      73.8151e351.0751e361.5064e361.2625e368.5286e351.7814e351.8552e351.9313e352.0097e352.0905e35
      87.0440e348.8342e349.3808e341.0586e351.6377e356.3515e346.6147e346.8861e347.1657e347.4537e34
      91.3534e361.3861e361.3839e361.2041e362.1576e362.3176e352.4137e352.5127e352.6147e352.7198e35
      104.8936e351.0126e361.1488e369.6788e351.2966e368.3978e348.7458e349.1046e349.4742e349.8551e34
      112.7691e359.9092e352.8587e352.9085e356.0377e351.6398e351.7078e351.7779e351.8501e351.9244e35
      123.7022e356.6988e356.2135e357.6659e356.8495e351.1526e351.2004e351.2496e351.3004e351.3526e35
      131.0913e351.6082e351.6359e352.1140e352.6405e357.3599e347.6649e347.9794e348.3034e348.6371e34

      Table 8.  The comparisons of the luminosity (L_{m}) due to RC effect by MMs with the observed values ( L_{\rm{rad}} ) for the five models I - V at n_{\rm B}=5.99\times10^{31}{\rm cm^{-3}}, m=10^{15}GeV, \sigma_m=10^{-24}{\rm cm^2}, \phi=10^{-26}{\rm cm^{-2}s^{-1}sr^{-1}}, T_6=10 ( \xi=4.9790\times10^{-15}\rm{cm^{-2}s^{-1}sr^{-1}} ).

      According to the analysis above, our results agree well with observations at lower relativistic densities and temperatures, but the greatest difference is about two orders of magnitude at higher relativistic densities and temperatures in WDs. The good agreement of our calculatedresults with observations shows that our model with the RC effect by MMs yields realistic results, as well as suggesting that the energy source of WDs is the RC effect by MMs.

      The monopole flux problem is well known to be a highly challenging and interesting issue. Some scholars have made pioneering work on this subject, such as Freese [32], Parker [39], Kolb & Turner [31], and Freese & Krasteva [40]. In this paper, we discuss this problem by considering the RC effect of MMs. Tables 10 ( m=10^{15}\rm{GeV}, \beta=9.4868\times10^{-3} ) and 11 ( m=10^{17}\rm{GeV}, \beta=1.00\times10^{-3} ) show the flux of MMs for the four models I \sim III and V, which correspond to typical asteroseismic correction factor values f_{\bigtriangleup\nu}=0.98, 0.99, 1.00, 1.02 when n_{\rm B}=5.99\times10^{31}, 1.89\times10^{32}\rm{cm^{-3}}. One can see that the maximum of the monopole flux values are 9.0935\times10^{-13}\rm{cm^{-2}s^{-1}sr^{-1}} and 5.8519\times10^{-13} \rm{cm^{-2}s^{-1}sr^{-1}} in Tables 10 and 11, respectively.

      star n_B=5.99\times10^{31}\rm{cm^{-3}} n_B=1.89\times10^{32}\rm{cm^{-3}}
      ξ(I)ξ(II)ξ(III)ξ(IV)ξ(V)ξ(I)ξ(II)ξ(III)ξ(IV)ξ(V)
      12.3509e-131.5233e-132.3050e-131.3813e-131.3978e-132.3614e-141.5301e-142.3152e-141.3875e-141.4041e-14
      21.9918e-136.1786e-145.6885e-148.3380e-146.7465e-142.0007e-146.2061e-155.7139e-158.3752e-156.7765e-15
      33.0933e-132.4021e-132.4794e-132.3920e-131.7884e-133.1070e-142.4128e-142.4904e-142.4026e-141.7964e-14
      48.0892e-148.5618e-146.3815e-146.8559e-145.2954e-148.1252e-158.5999e-156.4099e-156.8865e-155.3190e-15
      53.6993e-142.9033e-142.2223e-142.9015e-142.8746e-143.7157e-152.9162e-152.2322e-152.9145e-152.8874e-15
      69.0935e-136.5135e-137.0573e-136.0949e-137.8094e-139.1340e-146.5425e-147.0887e-147.0998e-147.8442e-14
      72.3248e-138.5916e-146.3833e-147.9257e-141.2204e-132.3352e-148.6299e-156.4117e-157.9610e-151.2259e-14
      84.4895e-133.7281e-133.6549e-133.3703e-132.2661e-134.5095e-143.7447e-143.6711e-143.3854e-142.2761e-14
      98.5262e-148.6703e-149.0402e-141.0812e-136.2765e-148.5642e-158.7090e-159.0804e-151.0860e-146.3044e-15
      108.5442e-144.3005e-143.9459e-144.8737e-143.7843e-148.5823e-154.3196e-153.9635e-154.8954e-153.8011e-15
      112.9486e-138.5810e-143.0965e-133.1671e-131.5870e-132.9617e-148.6192e-153.1103e-143.1812e-141.5940e-14
      121.5501e-138.9220e-141.0013e-138.4458e-149.8324e-141.5570e-148.9617e-151.0058e-148.4834e-159.8762e-15
      133.3578e-132.3731e-132.4285e-131.9557e-131.6286e-133.3727e-142.3837e-142.4394e-141.9644e-141.6359e-14

      Table 10.  The flux of MMs for the five models I−V (corresponding to typical asteroseismic correction factor values f_{\Delta\nu}= 0.98, 0.99, 1.00, 1.01, 1.02) when m=10^{15}{\rm GeV}, \beta=9.4868\times10^{-3}, n_{\rm B}=5.99\times10^{31}, 1.89\times10^{32}\rm{cm^{-3}}.

      starn_{\rm B}=5.99\times10^{31}\rm{cm^{-3} }n_{\rm B}=1.89\times10^{32}\rm{cm^{-3} }
      ξ(I)ξ(II)ξ(III)ξ(IV)ξ(V)ξ(I)ξ(II)ξ(III)ξ(IV)ξ(V)
      11.5302e-139.8450e-141.4791e-138.8011e-148.8428e-141.5370e-149.8889e-151.4857e-148.8403e-158.8822e-15
      21.3438e-134.1410e-143.7874e-145.5146e-144.4323e-141.3498e-144.1595e-153.8042e-155.5391e-154.4520e-15
      31.9655e-131.5150e-131.5522e-131.4864e-131.1031e-131.9742e-141.5218e-141.5591e-141.4930e-141.1080e-14
      45.5723e-145.8607e-144.3406e-144.6336e-143.5561e-145.5971e-155.8868e-154.3599e-154.6543e-153.5720e-15
      52.6798e-142.0914e-141.5918e-142.0667e-142.0359e-142.6918e-152.1008e-151.5989e-152.0759e-152.0450e-15
      64.9950e-133.5454e-133.8064e-135.8519e-134.1359e-135.0173e-143.5612e-143.8234e-145.8779e-144.1543e-14
      71.5209e-135.5812e-144.1175e-145.0764e-147.7618e-141.5276e-145.6060e-154.1359e-155.0990e-157.7963e-15
      82.5870e-132.1299e-132.0702e-131.8927e-131.2617e-132.5985e-142.1394e-142.0794e-141.9011e-141.2673e-14
      96.1421e-146.2105e-146.4385e-147.6561e-144.4190e-146.1695e-156.2382e-156.4672e-157.6902e-154.4387e-15
      105.8850e-142.9433e-142.6835e-143.2935e-142.5410e-145.9112e-152.9564e-152.6955e-153.3081e-152.5523e-15
      111.9284e-135.5727e-141.9968e-132.0279e-131.0090e-131.9370e-145.5975e-152.0057e-142.0370e-141.0135e-14
      129.8741e-145.6415e-146.2849e-145.2620e-146.0805e-149.9181e-155.6666e-156.3129e-155.2854e-156.1076e-15
      131.9890e-131.3942e-131.4150e-131.1301e-139.3331e-141.9979e-141.4004e-141.4213e-141.1351e-149.3746e-15

      Table 11.  The flux of MMs for the five models I−V (corresponding to typical asteroseismic correction factor values f_{\Delta\nu}= 0.98, 0.99, 1.00, 1.01, 1.02) when m=10^{17}{\rm GeV}, \beta=1.0\times10^{-3}, n_{\rm B}=5.99\times10^{31}, 1.89\times10^{32}\rm{cm^{-3}}.

      It is very interesting to note that the monopole flux decreases as n_{\rm B} increases from 5.99\times10^{31} to 1.89\times 10^{32}\rm{cm^{-3}} in Tables 10 and 11. This is not hard to understand according to Eq. (10). Based on our calculations above from Tables 10, due to the RC effect by MMs, we obtain new limits on the MM flux of \xi\leq 9.0935\times 10^{-13}\rm{cm^{-2}s^{-1}sr^{-1}} and \xi\leq 9.1340\times 10^{-14} \rm{cm^{-2}s^{-1}sr^{-1}} when n_{\rm B}=5.99\times10^{31} \rm{cm^{-3}} and 1.899\times 10^{32}\rm{cm^{-3}} , respectively. In Table 11, the new limits on the MM flux are \xi\leq 4.9950 \times10^{-13}\rm{cm^{-2}s^{-1}sr^{-1}} and \xi\leq 5.0173\times10^{-14} \rm{cm^{-2}s^{-1}sr^{-1}} when n_B=5.99\times10^{31} and 1.899\times10^{32} , respectively.

      Based on the above analysis, we obtain new limits on the MM flux of \xi\leq 9.0935\times10^{-13} \rm{cm^{-2}s^{-1}sr^{-1}}, and \xi\leq 4.9950\times10^{-13}\rm{cm^{-2}s^{-1}sr^{-1}} at n_{\rm B}=5.99\times10^{31}\rm{cm^{-3}} when m=10^{15}\rm{GeV}, \beta=9.4868\times10^{-3} , and m=10^{17}\rm{GeV}, \beta=10^{-3} , respectively. When we estimate the number of MMs captured, the MM luminosity and the limit of the MM flux, as samples, we consider the 13 RGB stars in our MM model for the following reasons. First, WDs originate from RGB stars. Second, compared with WDs, RGB stars have a very large surface area. According to Eq. (10), we expect RGB phases to capture more MMs during their evolution period. Third, since the MM is a superheavy particle, when MMs are captured by an RGB, they will be deposited in the star core. If all MMs captured by RGBs remain, their number will be much larger than that of those captured by WDs. Thus, the number of MMs calculated inside an WD will be more accurate than for MMs captured only during the WD phase. For example, Freese et al. [51] showed that if the MMs captured by stars in the main sequence stage all survive, the MM flow due to neutron star catalysis can be strengthened by up to 7 orders of magnitude. Finally, based on Schwarzschid [52], the nuclear energy generation rates of the proton-proton and CNO cycle are \epsilon_{\rm{pp}}\approx10\rho_{100}T_7^4 \rm{erg\; g^{-1}s^{-1}} and \epsilon_{\rm{CNO}}\approx8\rho_{100}T_7^{16} \rm{erg\; g^{-1}s^{-1}} , respectively (where T_7=T/10^7\rm{K} is the temperature, and \rho_{100}=\rho/100 is the density). Based on the discussions fof Bjork et al. [53], we may select the mass of the outer layer of the RGB as being from 0.005\sim0.02 {M}_{\odot} (the main component is hydrogen). Thus, when T_7=0.1-1, \rho_{100}\sim10^{-4} , and we obtain a proton-proton nuclear energy generation rate of 10^{24} - 10^{28} \rm{ergs\; s^{-1}}, which is \ll L_m=10^{34} - 10^{36}\rm{ergs\; s^{-1}} in our calculations. Based on the above analysis, and the fact that RGB stars are the origins of WDs, we therefore have L_{\rm{m}}\approx L_{\rm{rad}} .

      One can also conclude that with the increasing number of MMs captured, the luminosity due to the RC effect by MMs increases linearly with time until it becomes the main contribution to the total luminosity. One can even observe that for some of the oldest white dwarfs, the luminosity may have passed its minimum and some reheating may have occurred.

      It may be suggested that the annihilation of magnetic and antimagnetic monopoles could result in a significant reduction in the number of monopoles and the catalytic luminosity of the monopoles in the WDs. Dicus et al. [54] calculated the annihilation cross sections of magnetic monopoles and antimonopoles caused by two-body and three-body recombination. Their results show that this kind of annihilation has little effect on the flux and luminosity of the monopole. On the other hand, some WDs may have magnetic fields of up to 10^5 G according to observations. The forces generated by the magnetic field inside the white dwarf must balance the gravitational and Coulomb interactions.The magnetic field may keep the monopole and antimonopole distributions far enough apart for annihilation to be negligible.

      On the other hand, neutrino emission in WDs is a very interesting issue. Based on the discussions of Althaus et al. [55], when WD is very hot, neutrino emissions could be a major source of cooling. However, based on the relatively low temperature environment of WDs (e.g., T_6=1, 10 in our paper) we discussed the heating resource problem with our MMs model. The neutrino processes inside WDs at such low temperatures (e.g., T_6=1 ) may not be the main cooling process (see discussions by Itoh et al. [56]). On the other hand, Izawa [57] also discussed the neutrinos emitted according to Eqs. (23), (24) in his paper and calculated the neutrino emitted per one nucleon decay at low and high energy components, finding that these neutrino losses did not affect the structure or evolution of Rubakov stars because the energy lost through the neutrino emission is smaller than 100 MeV per one nucleon decayn although about two neutrinos are emitted furing the decay of one nucleon.

      According to the above analysis, one can see that MMs pass through space to be captured by WDs. MMs trapped inside a WD can catalyze the decay of nuclei, which can function as an energy source to keep the WDs hot.

    V.   CONCLUSIONS AND OUTLOOKS

    • We have presented five MMs models of WD energy resources to discuss their cooling based on certain observations of 13 RGB stars. We find that the number of MMs captured can reach a maximum value of 9.1223\times10^{24} when m=10^{17}\;\;{\rm GeV}, \; n_{\rm B}=5.99\times10^{31}\; \rm{cm^{-3}}, \phi=7.59\times10^{-26}\;\;\rm{cm^{-2}s^{-1}sr^{-1}}. The good agreement with observations of our luminosities due to the RC effect by MMs calculated for WDs shows that our model is reasonable. We conclude that the energy source of WDs may be the RC effect. Due to the RC effect by MMs, we obtain a new limit of MM flux of \xi\leq 9.0935\times 10^{-13} \;\; \rm{cm^{-2}s^{-1}sr^{-1}} and \xi\leq 4.9950\times10^{-13}\;\;\rm{cm^{-2}s^{-1}sr^{-1}} at n_{\rm B}=5.99\times 10^{31} \rm{cm^{-3}} when m=10^{15}\rm{GeV}, \;\beta=9.4868\times10^{-3}, and m=10^{17}\rm{GeV}, \;\beta=10^{-3}, respectively.

      In this paper, the main highlights may be given as follows. First, we created detailed estimates of the cooling ages of 13 RGB stars using the packages BASTA [42], isochrones [43], isoclassify [44], PARAM [45,46], and scaling-giants [47]. Second, we proposes five new models to discuss the energy resources and the cooling of WDs and compare the luminosities with observations for 13 RGB stars due to the RC effect. Finally, the new limit of the MM flux is obtained based on our models.

      As is widely known, research on MMs haa always been a hot frontier topic in the fields of nuclear physics and astrophysics. The search for MMs remians a difficult and challenging problem, and the flux of magnetic monopoles in the universe remains uncertain. The neutrino emissivity rates due to the RC effect also may play a key role in the process of WD and neutron star evolution. These challenging problem will be our future issues.

Reference (57)

目录

Link
IOPScience
SCOAP3
arXiv
Chinese Physical Society
Journal information
Introduction
Editorial Board
Others
Subscribe
Contact us
QQ: 1307685413
Tel: +86-010-88235947
Fax: +86-010-88233126
E-mail: cpc@ihep.ac.cn

  • CPC Website

  • IOP Website

  • CPC WeChat

Copyright © Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049, China 京ICP备05002789号-1

Supported by: Beijing Renhe Information Technology Co. Ltd  E-mail: info@rhhz.net

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return