As well known, white dwarfs (hereafter WDs) has no thermal nuclear burning in their Interior. The temperature in the interiors of WDs can be about 10⁶ K with total thermal energy less than 10⁴⁷ ergs. The radius of WDs can be about 10⁴ km with surface temperature $T\approx 3\times {10}^3~4\times {10}^4\text{K}$. From the Stefan-Boltzmann law, the radiation luminosity of WDs is

$L_{\text{rad}}=4\pi R^2\sigma T_{\text{eff}}^4\approx 7.1\times {10}^{30}{\left(\frac{R}{{10}^4\text{km}}\right)}^2{\left(\frac{T_{\text{eff}}}{{10}^4\text{K}}\right)}^4\mathrm{,}$(1)

where R is the radius of the star, and $\sigma =5.6704\times {10}^{-5}\text{ergs}\cdot {\text{s}}^{-\text{1}}\cdot {\text{cm}}^{-\text{2}}\cdot {\text{K}}^{-\text{4}}$ is the radiation constant from Stefan’s law. This surface temperature is defined in astronomy as the effective temperature $T_{\text{eff}}$ by means of Stefan’s law. For WDs with temperature 3 × 10³ - 4 × 10⁴ K, so that $L\approx 5.6\times {10}^{28}~1.8\times {10}^{33}\text{ergs}\cdot {\text{s}}^{-\text{1}}$.

The WDs energy sources problem has always been an interesting issue. Refs. [1] - [4] studied the WDs energy sources problem and showed that the burning of ²²Ne in WDs may be an extra heating source. However, Ref. [5] discussed this issue and showed that ²²Ne can not be a possible cause of the heating in WDs. In this paper, we present two models to solve this problem for 25 typical massive WDs selected from Ref. [6] . Our model is based on the magnetic monopoles (hereafter MMs) catalytic proton decay (RC effect) [7] [8] . The issues on MMs are forefront subjects in astrophysics (e.g., Refs. [9] - [13] ). We also studied the problem of MMs and other related issues (e.g., Refs. [14] - [20] ).

This paper is arranged as follows. In Section 2, we study the number of possible MMs captured by WDs, and the luminosity by RC effect. In Section 3, the MMs model and RC luminosity in WDs are discussed. Results and discussions are given in Section 4. We obtain some conclusions in Section 5.

The number of MMs captured in space at the surface of stars is given by follows [22]

$N_{m\left(\text{sur}\right)}=5.7\times {10}^{-25}{m}_9{v}_{-3}{n}_B\frac{{\zeta }_m^0}{{\zeta }_s}\mathrm{,}$(2)

where $v_{-3}=v_m/{10}^{-3}c$, $m_9=m_m/{10}^9{m}_p$, $v_m\mathrm{,}{m}_m$ are the velocity and the mass of the MMs, respectively. The MMs mass is about $m_m~{10}^{16}{m}_p$ [11] , and m_p c are the mass of the proton and the speed of light, respectively. ${\zeta }_m^0$ is the content of MMs in space. ${\zeta }_s\approx 1.9\times {10}^{-32}{m}_9$ is the maximum number of MMs, which is defined as Newton saturation value. $n_B=\left({10}^4~{10}^6\right){\text{cm}}^{-3}$ is the number density of baryons in the space of the Milky Way galaxy [22] . In Equation (2), the number density of nucleons can be written by [22]

$n_B=2.90\times {10}^{16}{\left(\frac{R}{R_g}\right)}^{-3}{\left(M_{12}\right)}^{-2}=2.242\times {10}^{24}{M}_{\mathrm{<mo>\; *\; </mo>}}{R}_{\mathrm{<mo>\; *\; </mo>}}^{-3}{\text{cm}}^{-3}\mathrm{,}$(3)

where $R_g=2.96\times {10}^5{M}_{\mathrm{<mo>\; *\; </mo>}}$ is the Schwarzschild radii, and $M_{12}=M_{\mathrm{<mo>\; *\; </mo>}}/{10}^{12}$, $M_{\mathrm{<mo>\; *\; </mo>}}=M/M_{\odot }$, and $R_{\mathrm{<mo>\; *\; </mo>}}=R/R_{\odot }$.

According Equations (2~3), the total number of MMs trapped in space after the formation of stars (or planets) is estimated to be

$\begin{array}{c}{N}_{m\left(\text{tot}\right)}=4\pi R^{2}t{N}_{m\left(\text{sur}\right)}=3.463\times {10}^7{v}_{-3}{n}_B{t}_9{m}_9\left(\frac{{\zeta }_m^0}{{\zeta }_s}\right){\left(\frac{R}{R_{\odot }}\right)}^2\\ =3.463\times {10}^7\xi v_{-3}{n}_B{t}_9{R}_{\ast }^2\mathrm{,}\end{array}$(4)

where $R_{\ast }=R/R_{\odot }$, $t_9=t/{10}^9\text{yr}$, and $\xi =m_{9}F/{\zeta }_s$. R, t, and F are the radius, the age of the star, and the MMs flux in space, respectively.

$pM\to e^{+}{\pi }^{0}M+\text{debris}\left(\mathrm{85\; <mi>\; \%\; </mi>}\right)$, and $pM\to e^{+}{\mu }^{\pm }M+\text{debris}\left(\mathrm{15\; <mi>\; \%\; </mi>}\right)$ are named proton decay catalyzed by MM, which proposed by Callen and Rubacov (RC effect) [7] [8] . The luminosity due to the RC effect is [22]

$L_m\approx \frac{4\pi }{3}{r}_c^3{n}_m{n}_B\langle {\sigma }_m{v}_{\text{T}}\rangle m_B{c}^2=N_m{n}_B\langle {\sigma }_m{v}_{\text{T}}\rangle m_B{c}^2\mathrm{,}$(5)

where $v_{\text{T}}=\sqrt{kT/m_B}\approx 8.691\times {10}^3{T}^{1/2}\text{cm}/\text{s}$ is the thermal movement speed of the nucleus relative to the MM. n_m T and k are the number density of MMs, the temperature and the Boltzmann constant, respectively. $m_B\approx 1.78\times {10}^{-24}\text{g}$ and $r_c$, are the nucleons mass and the radius of the stellar central region, respectively. ${\sigma }_m$ is the reaction cross section, whose range is about 10⁻²⁶ - 10⁻²⁴ cm². By using the SU(5) grand unification theory, Ref. [23] gave the value of ${\sigma }_m\approx 4.28676\times {10}^{-24}$.

The MMs flux has been considerable interest issue. Parker (1970) gave the MMs flux as $F\le {10}^{-16}{\text{cm}}^{-\text{2}}\cdot {\text{s}}^{-\text{1}}\cdot {\text{sr}}^{-\text{1}}$ [21] . Ref. [26] gave a limit on the flux by $F{\left(\sigma v\right)}_{-28}\le {10}^{-21}{\text{cm}}^{-\text{2}}\cdot {\text{s}}^{-\text{1}}\cdot {\text{sr}}^{-\text{1}}$ in neutron star. Ref. [27] discussed the MMs flux, which may be $F{\left(\sigma v\right)}_{-28}\le 2\times {10}^{-18}{\text{cm}}^{-\text{2}}\cdot {\text{s}}^{-\text{1}}\cdot {\text{sr}}^{-\text{1}}$ in WDs. Ref. [28] also shown that the bound was stated as $F{\left(\sigma v\right)}_{-28}\le {10}^{-28}{\text{cm}}^{-\text{2}}\cdot {\text{s}}^{-\text{1}}\cdot {\text{sr}}^{-\text{1}}$. In this paper, we select the MMs flux of $F=1.9\times {10}^{-23}{\text{cm}}^{-\text{2}}\cdot {\text{s}}^{-\text{1}}\cdot {\text{sr}}^{-\text{1}}$ and the other parameters are selected as $n_B^0={10}^5{\text{cm}}^{-3}$, $T_6=\mathrm{0.01,0.1,1,10}$, $m_m={10}^{16}{m}_p$, $\xi ={10}^2$. We selected 25 typical high mass WDs from [6] . Some main parameters can be reference in Table 3 of Ref. [6] . Figure 1 and Figure 2 show the number of magnetic monopoles captured from space in the lifetime of the O + Ne(C + O) core high mass WDs for model (I), and (II) as a function of $M_{\mathrm{<mo>\; *\; </mo>}}$ when $\xi ={10}^2$, and $n_B^0={10}^6{\text{cm}}^{-3}$ at $T_6=\mathrm{0.01,0.1,1,10}$. As the temperature increases, the number of MMs captured increases by about two, and three orders of magnitude for model (I), and (II), respectively. However, the number of MMs captured increases by about three orders of magnitude for model (II).

When the temperature is certain, the number of MMs captured can be $N_2>N_1$ and the higher the mass, the larger the number of MMs captured becomes from Figure 1 . It is because that according to Equations (6) (8) (10) (11), we know that

$N_1\propto {\left(1/M_{\mathrm{<mo>\; *\; </mo>}}\right)}^{2/3}$, but $N_2=\exp \left[\frac{\ln \left(1.36719\times {10}^{31}F{t}_9{M}_{\mathrm{<mo>\; *\; </mo>}}^{-1.35}{v}_{-3}^{-2}\right)}{0.55}\right]$. The effect of mass

radius relation on the number of the MM captured is ignored in model (I), Model (II) considers the effect of mass radius relation and RC effect on the number of captured monopoles, so the data is relatively accurate.

In WDs, monopoles captured can catalyze proton decay and provide the internal heating. A monopole which passes through WDs can also lose enough energy. Ahlen and Kinoshita (1982) calculated these energy, which given by $\text{d}E/\text{d}x\approx 100\rho \beta \text{GeV}/\text{cm}$, where $\rho$ is the density of the WDs (in g·cm⁻³), and $\beta$ is the velocity of the MM as it passes through the WDs [29] . The energy loss in traveling through the WDs can be about 5 × 10¹⁷ MeV. If the MM is captured, it sinks toward the center of the WDs and the time scale for the MM to fall from rest to the center can be estimated to be about 1000 s.

Figure 2 display that the luminosities as a function of $M_{\mathrm{<mo>\; *\; </mo>}}$ for WDs at the temperature of $T_6=\mathrm{0.01,0.1,1,10}$ when $\xi ={10}^2$, $n_B^0={10}^6{\text{cm}}^{-3}$. The luminosities increase as the temperature increases. By comparing the luminosities of the observations with those of model (I), and (II), we find that the luminosities of model (II) are agreed well with those of the observations at relativistic high temperature (e.g., $T_6=\mathrm{1,10}$), However, the luminosities of the observations can be five orders of magnitude larger than those of model (I).

Based on the above analysis, one can conclude that with the increasing of the number of MMs captured by the WDs, the luminosity of MMs catalyzed nuclear decay increases linearly with time until it becomes the main contribution to the total luminosity. Even one can observe that for some of the oldest white dwarfs, the luminosity may have passed its minimum, and some reheating may have occurred. The annihilation of MM and anti-MM may make a significant reduction in the number of MMs and the catalytic luminosity of the monopole in the WDs. Ref. [30] calculated the annihilation cross sections of MMs and anti-MMs caused by two-body and three-body recombination. Their results showed that the annihilation has little effect on the flux and luminosity.

Based on the MMs catalytic proton decay, we present two MMs models of the energy resources in WDs. We discuss the luminosity to apply to 25 massive WDs and calculated the number of MMs captured by WDs. We also compare these luminosity of the two MMS model with the observations. We find that the luminosities increase as the temperature increases. The luminosities for most of massive WDs for model (II) are agree well with the observations and the difference is no more than one order of magnitude at relativistic high temperature (e.g., $T_6=\mathrm{1,10}$). However, the luminosities of the observations can be five orders of magnitude larger than those of model (I). According to our calculations and discussion, the monopole-catalyzed proton decay process may be an effective way, which can prevent WDs from cooling.

This work was supported in part by the National Natural Science Foundation of China under grants 11965010, 11565020, and the Natural Science Foundation of Hainan Province under grant 2019RC239, 118MS071, 114012 and the Counterpart Foundation of Sanya under grant 2016PT43, 2019PT76, the Special Foundation of Science and Technology Cooperation for Advanced Academy and Regional of Sanya under grant 2016YD28, the Scientific Research Starting Foundation for 515 Talented Project of Hainan Tropical Ocean University under grant RHDRC201701.