Possibility of Searching for Accreting White Dwarfs with the Chinese Space Station Telescope

With the assumption of optically thick wind, Hachisu et al. (1999) studied the evolution of accreting WDs with different WD masses and accretion rates. They presented the dependence of radius and effective temperature on the accretion rate for different WD masses (see their Figures 3 and 4). For any given WD mass and accretion rate, we can get the radius and effective temperature by interpolation with these data. Then we can get the surface gravity of an accreting WD with a given WD mass and radius. In Figure 1, we show the surface gravities and effective temperatures of accreting WDs with different WD masses. From this plot, we can find that there is a rough linear relation between the logarithm of effective temperature and logarithm of surface gravity. This rough linear relation can be understood as follows. From previous studies (e.g., Chen et al. 2019a), we know that the evolution of accreting WDs mainly depends on their accretion rates. From Hachisu et al. (1999), we can find that there is a rough linear relation between the logarithm of accretion rate and the logarithm of effective temperature or photospheric radius (see their Figures 3 and 4). Then we can fit these data with Equation (1). The coefficients for different WD masses are given in Table 1.

$$\begin{eqnarray}&&\mathrm{log}g={k}_{i}\mathrm{log}{T}_{\mathrm{eff}}+{c}_{i}\end{eqnarray} \tag{ 1 }$$

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Regarding the spectra of accreting WDs, we adopt the spectra computed by Rauch et al. (2018). ³ They have adopted the non-local thermodynamic equilibrium stellar atmosphere model to compute grids of spectra for WD models with metallicity Z = 0.02, 0.001 and pure H. These spectra have been successfully applied to explain the spectra of novae and supersoft X-ray sources (Rauch et al. 2010; Rauch 2011). In Figure 2, we present an example of the spectrum for an accreting WD with $\mathrm{log}(g\,{\mathrm{cm}}^{-1}\,{{\rm{s}}}^{2})=4.0$, T_eff = 58,000 K and pure H. Because the grid of spectra for WD models with pure H is much denser than models with solar metallicity, we adopt the grid of spectra for WD models with pure H in our calculation. Moreover, we have tested other choices of spectra grid and find that it does not significantly influence our results. There are some parameters of accreting WDs, which are not covered by this grid. We use the blackbody spectra for these accreting WDs.

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In order to get the magnitudes of accreting WD binaries, we also need consider the emission from the companion stars of the accreting WDs. The typical companions of accreting WDs are MS, HG and RG stars. ⁴ For a given WD mass, we constrain its companion mass as follows. The upper limit of the companion star's mass will be the critical mass at which the binary system still undergoes stable mass transfer. Following Hurley et al. (2002) and Han et al. (2002), we adopt the critical mass ratio, 4.0, 5.0 and 1.5 for binaries with MS, HG, RG companions, respectively. On the other hand, if the donor mass is too small, the mass transfer rate will be smaller than the lower limit of stable burning rate. Then the system will not evolve into an SNBWD or RAWD. Therefore, we chose these systems with thermal timescale mass transfer rates larger than the minimum stable burning rate. The thermal timescale mass transfer rate (${\dot{M}}_{\mathrm{th}}$) can be estimated with the following equation.

$$\begin{eqnarray}&&{\dot{M}}_{\mathrm{th}}=\displaystyle \frac{M}{{\tau }_{\mathrm{th}}}=\displaystyle \frac{2{RL}}{{GM}},\end{eqnarray} \tag{ 2 }$$

where M, R, L are the mass, radius and luminosity of the donor star, respectively. G is the gravitational constant. τ_th is the thermal timescale of the donor star (Kippenhahn et al. 2012). The minimum stable burning rate for a given WD mass can be given with the following equation (Chen et al. 2019a).

$$\begin{eqnarray}&&\begin{array}{l}\qquad \mathrm{log}{\dot{M}}_{\mathrm{low}}=-10.35+8.37\cdot {M}_{\mathrm{WD}}-6.84\\ \qquad \,\cdot {M}_{\mathrm{WD}}^{2}+2.07\cdot {M}_{\mathrm{WD}}^{3},\end{array}\end{eqnarray} \tag{ 3 }$$

where M_WD is the WD mass in units of solar mass and ${\dot{M}}_{\mathrm{low}}$ is in units of M_⊙ yr⁻¹.

For a donor star with a given mass, we can get its effective temperature and surface gravity from the Modules for Experiments in Stellar Astrophysics isochrones and evolutionary tracks (MIST; Choi et al. 2016; Dotter 2016). In MIST, different evolutionary phases are separated. But HG and RG phases are not separated. Following Hurley et al. (2000), we separated HG and RG phases according to the stellar radius. If the radius is larger than the radius at the end of HG phase, then the star is on RG phase, otherwise on HG phase. The radius at the end of HG phase for a given mass can be calculated with the formulas in Hurley et al. (2000).

Regarding the spectra of the companions, we make use of the database ybc, ⁵ which provides different types of stellar libraries for stars with different types (Chen et al. 2019b). For any companion star, we can select its spectra from this database according to its effective temperature and surface chemical abundances following the selection scheme in their Section 3.10.

For a source with observed flux f_λ , its magnitude on any i band with transmission S_λ,i is given by

$$\begin{eqnarray}&&{m}_{i}=-2.5\mathrm{log}\left[\displaystyle \frac{{\int }_{{\lambda }_{1}}^{{\lambda }_{2}}\lambda {f}_{\lambda }{S}_{\lambda ,i}d\lambda }{{\int }_{{\lambda }_{1}}^{{\lambda }_{2}}\lambda {f}_{\lambda }^{0}{S}_{\lambda ,i}d\lambda }\right],\end{eqnarray} \tag{ 4 }$$

where ${f}_{\lambda }^{0}$ is the flux of the reference object. λ₁ and λ₂ are the filter's maximum and minimum wavelength. For the CSST, AB magnitude is adopted as the reference system. The flux of reference is ${f}_{\nu }^{0}={10}^{\tfrac{48.60}{-2.5}}\mathrm{erg}\,{{\rm{s}}}^{-1}\,{\mathrm{cm}}^{-2}\,{\mathrm{Hz}}^{-1}$ (Chen et al. 2019b) and ${f}_{\lambda }^{0}=\tfrac{c}{{\lambda }^{2}}{f}_{\nu }^{0}$ (c is the speed of light in a vacuum).

The absolute magnitude of the source on i band is

$$\begin{eqnarray}&&{M}_{i}=-2.5\mathrm{log}\left[{\left(\displaystyle \frac{R}{10\mathrm{pc}}\right)}^{2}\displaystyle \frac{{\int }_{{\lambda }_{1}}^{{\lambda }_{2}}\lambda {F}_{\lambda }{10}^{-0.4{A}_{\lambda }}{S}_{\lambda ,i}d\lambda }{{\int }_{{\lambda }_{1}}^{{\lambda }_{2}}\lambda {f}_{\lambda }^{0}{S}_{\lambda ,i}d\lambda }\right],\end{eqnarray} \tag{ 5 }$$

where R is the radius of the source, A_λ is the extinction and F_λ is the intrinsic flux (Chen et al. 2019b). In this paper, we assume that the extinction is zero.

Given that the spatial resolution of the CSST is 0.15'', we can resolve a binary at 1 kpc if the binary separation is larger than 150 au, i.e., 3.2 × 10⁴ R_⊙. Therefore, we would expect that most binaries are unresolvable in the survey of the CSST. For an unresolved accreting WD binary with two components, the magnitude of the binary system can be given by the following equation.

$$\begin{eqnarray}\begin{array}{rcl}m & = & -2.5\mathrm{log}\left[\displaystyle \frac{{\displaystyle \int }_{{\lambda }_{1}}^{{\lambda }_{2}}\lambda {f}_{\lambda 1}{S}_{\lambda ,i}d\lambda +{\displaystyle \int }_{{\lambda }_{1}}^{{\lambda }_{2}}\lambda {f}_{\lambda 2}{S}_{\lambda ,i}d\lambda }{{\displaystyle \int }_{{\lambda }_{1}}^{{\lambda }_{2}}\lambda {f}_{\lambda }^{0}{S}_{\lambda ,i}d\lambda }\right]\\ & = & {m}_{1,i}-2.5\times \mathrm{log}\left(1+{10}^{\tfrac{{m}_{1,i}-{m}_{2,i}}{2.5}}\right),\end{array}\end{eqnarray} \tag{ 6 }$$

where f_λ1, f_λ2 are the observed fluxes of the two components of the binary system, respectively. m_1,i and m_2,i are the magnitudes of the two components on the i band, respectively. In a similar way, we can get the absolute magnitude of the unresolved binary with the absolute magnitudes of two components.

In Figure 3, we show the location of accreting WDs in the Kiel and HR diagrams. From this plot, we can find that SNBWDs have higher effective temperatures and larger surface gravities compared with RAWDs. This can be understood as follows. Because SNBWDs and RAWDs have comparable H burning rates, they have comparable luminosities. For RAWDs, optically thick wind occurs (Kato & Hachisu 1994). This leads to the modest expansion of the accreting WD's photospheric radius, lowering their effective temperatures.

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In order to compare these accreting WDs with other types of stars, we also plot a grid of isochrones with ages from $\mathrm{log}(t/\mathrm{yr})=6.6\mbox{--}10.1$ in 0.05 dex steps from the mist (version 1.2) database (Choi et al. 2016; Dotter 2016). The initial stellar masses in these isochrones range from 0.1 M_⊙ to 73 M_⊙. Although the initial stellar mass can be larger than 73 M_⊙, we have checked that the upper limit of the mass range has little influence of our conclusions. The initial metallicity is assumed to be solar metallicity and the initial rotation velocity is v/v_crit = 0.40 (v_crit is the critical surface velocity). From these two plots, we can find that these accreting WDs are much brighter than these isolated WDs which have $\mathrm{log}(L/{L}_{\odot })$ ∼(−4) −4 and $\mathrm{log}({T}_{\mathrm{eff}}/{\rm{K}})\sim 3.5\mbox{--}5.5$ and are located on the cooling sequence. Their parameters are close to some post AGB stars and the central stars of planetary nebulae. These SNBWDs have relatively larger effective temperatures and surface gravities in contrast with RAWDs. In Figure 4, we also show the location of the companions of accreting WDs in the Kiel and HR diagrams. From this plot, we can find that the accreting WDs are much brighter and have higher effective temperatures compared with their companions. From Equation (6), we can find that the magnitudes of the unresolved accreting WD binary systems will be very close to the magnitudes of the accreting WDs.

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In order to know if we can search for accreting WDs with the CSST, we try to use different combinations of colors of the CSST and find the possible color–color diagrams to distinguish accreting WD binaries from other stars with different types. In Figure 5, we show the (NUV − y, u − y) color–color diagram for accreting WD binaries. In addition, we also show other types of stars in the same plot. Given that the possible observable errors of these colors are smaller than 0.1 dex, we can find that accreting WD binaries with NUV − y ∼ −2.5 and u − y ∼ −2.1 are distinguishable from stars with other types from this plot. The accreting WDs in these binaries have effective temperatures $\mathrm{log}({T}_{\mathrm{eff}}/{\rm{K}})\gt 5.0$ and $\mathrm{log}(g\,{\mathrm{cm}}^{-1}\,{{\rm{s}}}^{2})\gt 5.0$. Their WD masses range from 0.60 M_⊙ to 1.377 M_⊙. The accretion rates of these accreting WDs are close to the accretion rate in the stable burning regime and smaller than 10⁻⁵ M_⊙ yr⁻¹. But for other accreting WDs, we cannot distinguish with this color–color diagram. In Figure 6, we show the (NUV − r, u − g), (NUV − i, u − g), (NUV − z, u − g) and (NUV − y, u − g) color–color diagrams for accreting WD binaries. We can find that some accreting WD binaries with colors ∼−0.5−0 can be distinguished from other stars. These accreting WDs are RAWDs. They have effective temperatures around 10⁴ K and accretion rates of 10⁻⁵–10⁻⁴ M_⊙ yr⁻¹.

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In Figure 7, we show the (NUV − i, u − g) color–color diagram for accreting WDs in a simple stellar population with an age around ${\mathrm{log}}_{10}(t/\mathrm{yr})=10\mbox{--}10.1\,$ Given the old age of this population, only these post-AGB stars have colors close to the accreting WD binaries. From this plot, we can find that most accreting WDs may be easy to be found in this simple stellar population, in particular these accreting WDs with NUV − i ranging from −1.0 to −1.5 (see Figure 6). These results indicate that globular clusters may be a good place to search for accreting WD binaries.

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In our work, we do not consider the impact of interstellar medium on the spectra of the accreting WDs. Given the high temperatures of accreting WDs, their emission should be capable of ionizing the interstellar medium, producing some emission lines, such as He ii λ4686, $\left[{\rm{O}}\,{\rm\small{III}}\right]\lambda 5007$, and [O i] λ6300 (Rappaport et al. 1994). This may also produce a nebulae around the accreting WD, similar to CAL 83 (Remillard et al. 1995). But Woods & Gilfanov (2016) found that whether an accreting WD has a nebular around it will depend on the density of interstellar medium. If the density of interstellar medium is too low, there will be no nebular around the accreting WD. Moreover, they demonstrated that most supersoft X-ray sources are in low density media. Therefore, we would expect that the interstellar medium might have little impact on our results.