Two-temperature Radiative Hot Accretion Flow around Neutron Stars

The ions and electrons interchange energy through Coulomb collision (q^ie). The formula for q^ie is given by Narayan & Yi (1995), hereafter NY95). The electrons can cool by bremsstrahlung radiation (${q}_{\mathrm{brem}}^{-}$), synchrotron radiation (${q}_{\mathrm{syn}}^{-}$), the self-Comptonization of bremsstrahlung radiation (${q}_{\mathrm{brem},\ {\rm{C}}}^{-}$), and the self-Comptonization of synchrotron radiation (${q}_{\mathrm{syn},\ {\rm{C}}}^{-}$). Therefore, the electron cooling term in Equation (4) can be written as ${Q}^{-}={q}_{\mathrm{brem}}^{-}+{q}_{\mathrm{syn}}^{-}+{q}_{\mathrm{brem},\ {\rm{C}}}^{-}+{q}_{\mathrm{syn},\ {\rm{C}}}^{-}$. The expressions for (${q}_{\mathrm{brem}}^{-}$), (${q}_{\mathrm{syn}}^{-}$), (${q}_{\mathrm{brem},\ {\rm{C}}}^{-}$) and (${q}_{\mathrm{syn},\ {\rm{C}}}^{-}$) are given in NY95. As in Bu & Gan (2018), we use a Compton enhancement factor η to calculate the terms ${q}_{\mathrm{brem},\ {\rm{C}}}^{-}$ and ${q}_{\mathrm{syn},\ {\rm{C}}}^{-}$. Note that the optical depth needed for calculating η at any radii is calculated from $\theta =0$ to π.

There are complicated interactions between the hard surface of a NS and the accretion flow. The interactions are hot topics and quite complicated. For example, when an accretion flow approaches the surface of a NS, the rotational velocity of the accretion flow should be decreased to the value equal to that of the NS. How to decrease the rotational velocity of the accretion flow is a hot topic (Popham & Narayan 1992; Narayan & Popham 1993; Inogamov & Sunyaev 1999; Medvedev & Narayan 2001; Popham & Sunyaev 2001; Medvedev 2004; Gilfanov & Sunyaev 2014; Burke et al. 2018). In the present paper, we simplify the interactions between a HAF and the surface of a NS. We just assume that when a HAF arrives at the surface of a NS, all of the energy carried by the HAF will be thermalized as blackbody emission and radiated out spherically (see Qiao & Liu 2018; Paper I). The thermal photons go outwards through the HAF and cool the HAF via Comptonization. The energy flux carried by the accretion flow onto the surface of a NS can be expressed as,

$$\begin{eqnarray}&&{L}_{* }=2\pi {R}_{* }^{2}{\int }_{0}^{\pi }{\rho }_{* }\min [{v}_{r* },0]\left[\displaystyle \frac{1}{2}{v}_{* }^{2}+{e}_{* }/{\rho }_{* }\right]\sin \theta d\theta .\end{eqnarray} \tag{ 7 }$$

Where, ρ_*, v_r*, v_* and e_* are density, radial velocity, total velocity, and internal energy at the surface of a NS (R_*), respectively. The radius for a NS with ${M}_{\mathrm{NS}}=1.4{M}_{\odot }$ is ∼12.5 km ∼ 3r_s. In this paper, we set R_* = 3r_s. In this paper, the inner boundary of our simulations is located at 3r_s. Therefore, we can set the physical variables at the surface of a NS to be same as those at the inner boundary. We further assume that the blackbody emission photons at the surface of a NS are distributed spherically. The effective temperature (T_*) of the blackbody emission at the surface of a NS can be expressed as,

$$\begin{eqnarray}&&{T}_{* }={\left(\displaystyle \frac{{L}_{* }}{4\pi {R}_{* }^{2}\sigma }\right)}^{1/4}\end{eqnarray} \tag{ 8 }$$

with σ being the Stefan–Boltzmann constant. The soft photons generated at the surface of a NS can move outward radially and cool the accretion flow via inverse-Comptonization. The inverse-Compton cooling rate in Equation (4) can be expressed as follows (Sazonov et al. 2005),

$$\begin{eqnarray}&&{Sc}=4.1\times {10}^{-35}{n}^{2}({T}_{* }-{T}_{{\rm{e}}})\xi \end{eqnarray} \tag{ 9 }$$

with n and T_e being the number density of electrons and temperature of electrons, respectively. We define ϱ = 0.5 and m_p to be the mean molecular weight and the proton mass, respectively. The number density of electrons can be expressed as n = ρ/ϱm_p. $\xi ={L}_{* }{e}^{-{\tau }_{{es}}(r)}/{{nr}}^{2}$ is the ionization parameter, with ${\tau }_{{es}}(r)={\int }_{0}^{r}\rho {\kappa }_{{\rm{X}}}{dr}$ being the X-ray scattering optical depth in radial direction. We set ${\kappa }_{{\rm{X}}}=0.4\,{\mathrm{cm}}^{2}\,{{\rm{g}}}^{-1}$.

For the accretion flow around a BH, we set Sc = 0.

The inner radial boundary of our simulations is located at 3r_s. The outer radial boundary is located at 1000r_s. As introduced above, for the simulations of HAFs around a BH, we set ${r}_{s}=2{{GM}}_{\mathrm{BH}}/{c}^{2}$. For the simulations of HAFs around a NS, we set r_s = 2GM_NS/c². Our simulation boundaries in θ direction are located at θ = 0 and θ = π. We use 192 × 88 grids to resolve the computational domain. To have higher resolution close to the compact object, in the r direction, the grids are logarithmically spaced. In θ direction, we put the grids uniformly. At the inner and outer radial boundary, we use outflow boundary conditions. At the rotational axis, we use axis-symmetry boundary conditions.

The initial condition is a rotating torus with constant specific angular momentum embedded in a nonrotating low-density medium. The exact formula for the torus can be found in Papaloizou & Pringle (1984, see also Stone et al. 1999). The torus center (or density maximum) is located at 300r_s. The maximum density at the torus center is ρ_max (see Table 1 for the values of ρ_max in different models).

Table 1.  Simulation Parameters and Results

Models	Central Object	ρmax	Compton Temperature T*	Mass Inflow Rate at
		(${10}^{-8}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$)		Inner Boundary (${\dot{M}}_{\mathrm{in}}({r}_{\mathrm{in}})/{\dot{M}}_{\mathrm{Edd}}$)
(1)	(2)	(3)	(4)	(5)
NS1	Neutron star	3	7.1 × 106 K	1.1 × 10−2
NS2	Neutron star	1	5.3 × 106 K	3 × 10−3
NS3	Neutron star	0.3	2.6 × 106 K	1.6 × 10−4
BH1	Black hole	26	⋯	1 × 10−2
BH2	Black hole	13	⋯	3.7 × 10−3
BH3	Black hole	2	⋯	1.7 × 10−4

Note. Col. 1: model names. Col. 3: the maximum density at the initial torus center in unit of ${10}^{-8}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$. Col. 4: Compton temperature of photons emitted at the surface of the NS (Equation (8)). Col. 5: time-averaged mass accretion rate of the central object (in unit of Eddington rate).

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In the accretion flow around a NS, one portion of the viscous heating energy (Q_vis) is stored in gas as entropy and advected to the NS. The second portion of the viscous heating energy is lost by radiation of electrons (Q⁻). The last portion of the viscous heating energy is lost through the inverse-Compton scattering cooling (the term S_c in Equation (4)) via collision between the soft photons from the surface of the NS and the accretion flow. We define the ratio of Q⁻to the viscous heating rate as

$$\begin{eqnarray}&&{f}_{c}=\displaystyle \frac{{Q}^{-}}{{Q}_{\mathrm{vis}}}.\end{eqnarray} \tag{ 13 }$$

We define the ratio of S_c to the viscous heating rate as

$$\begin{eqnarray}&&{f}_{N}=\displaystyle \frac{{S}_{c}}{{Q}_{\mathrm{vis}}}.\end{eqnarray} \tag{ 14 }$$

The radial profiles of f_c and f_N are shown in the bottom-left and bottom-right panels of Figure 1, respectively. Compared with models NS1 and NS2, the mass accretion rate in model NS3 is lowest (see the top left panel of Figure 1). In this model, the accretion rate is low, radiative cooling is not important. The cooling term (Q⁻) is more than one order of magnitude smaller than the viscous heating term. Although most of the viscous heating energy is advected to the surface of the NS and emitted out there, the emitted thermal soft photons have very low interaction efficiency with the accretion flow (f_N ≪ 1). The reason is as follows. The inverse-Compton cooling (Equation (9)) ${S}_{c}\propto {n}^{2}{L}_{* }/n\propto {n}^{2}\dot{M}/n$. $\dot{M}$ is proportional to n. Therefore, we have S_c ∝ n². In model NS3, the very low-density results in a very low interaction efficiency between soft photons from the surface of the NS and the accretion flow. The dynamical properties of the HAF in model NS3 are quite similar as those of a nonradiative HAF around a BH. For example, the mass inflow rate decreases inwards (see the top left panel of Figure 1). The mass inflow rate in the region 10r_s < r < 100r_s can be described as a power-law function of radius ${\dot{M}}_{\mathrm{in}}\propto {r}^{s}$, with s = 0.57. Numerical simulations of nonradiative HAFs around BHs also find that mass inflow rate outside 10r_s can be described as a power-law function of radius, with the power-law index in the range 0.5 < s < 1 (e.g., Stone et al. 1999; Yuan et al. 2012, 2015; Bu et al. 2013; Sadowski et al. 2013). As explained in previous works (e.g., Yuan et al. 2012, 2015; Bu et al. 2013; Sadowski et al. 2013; Gu 2015; Almeida & Nemmen 2019), the inward decrease of mass inflow rate is due to the presence of outflow/wind (green dashed line in top left panel of Figure 1). The presence of wind generated by a HAF has been confirmed by observations (e.g., Crenshaw & Kraemer 2012; Wang et al. 2013; Cheung et al. 2016; Homan et al. 2016; Almeida et al. 2018; Ma et al. 2019; Park et al. 2019). If the accretion rate is smaller than that in model NS3, the values of f_c and f_N will be smaller, and the dynamical properties of the HAF around a NS will be same as those of a nonradiative HAF around a BH.

In model NS2, the accretion rate (or density) is higher than that in model NS3. For HAF, viscous heating rate is proportional to n (see the second term on the right-hand side of Equation (3)). With the increase of accretion rate (or density), the radiative cooling rate increases faster than viscous heating rate. For example, the bremsstrahlung cooling rate ${q}_{\mathrm{brem}}^{-}\propto {n}^{2}$. With the increase of accretion rate, the inverse-Compton cooling (S_c) rate also increases faster than viscous heating rate because S_c ∝ n². From the bottom-left panel, we can see the value of f_c in model NS2 is much bigger than that in model NS3. However, in model NS2, f_c is still smaller than 1. The value of f_N is smaller than 1 inside 15r_s. Outside 15r_s, f_N > 1. In model NS2, the inverse-Compton cooling rate due to the presence of the thermal soft photons from the surface of the NS dominates viscous heating rate outside 15r_s. In model NS1, the value of f_c in the region 3.2r_s < r < 10r_s is significantly larger than 1. In the region 10r_s < r < 20r_s, the value of f_c can also be larger than 1. Outside 20r_s, the value of f_N is much larger than 1. Therefore, in model NS1, almost at the whole radii, the cooling rate is larger than the viscous heating rate. The parameter ${f}_{N}={S}_{c}/{Q}_{\mathrm{vis}}\propto {n}^{2}\exp (-{\tau }_{{es}}(r))/n\propto n\exp (-{\tau }_{{es}}(r))$. The density-weighted radial profiles of τ_es(r) are shown in the left panel of the third row. In the region r < 10r_s, the values of τ_es(r) in all models are much smaller than 1. Therefore, the value of f_N in this region mainly depends on n. In the region r < 10r_s, the value of f_N in model NS1 is much bigger than that in model NS2. Outside the region 10r_s, the electron scattering optical depth in radial direction (τ_es(r)) in model NS1 is significantly larger than that in model NS2. The larger value of τ_es(r) in the region outside 10r_s in model NS1 (compared with that in model NS2) makes the value of f_N in this region much smaller than that in the same region in model NS2.

In model NS1, there is a sharp jump of f_c at ∼10r_s. The sharp jump is not a numerical noise. Yuan (2001) analytically studied the properties of HAF. In his highest accretion rate model, it is found that outside 10r_s, the value of f_c increases inwards. It is also found that outside 10r_s, f_c is ∼1. A sharp jump of f_c at r ∼ 10r_s is also found (see the right panel of Figure 2 in Yuan 2001). The sharp jump behavior of f_c in models NS1 and BH1 (see the bottom-left panel of Figure 4) is quite similar as that found by Yuan (2001). The physical reason for the sharp jump of f_c at r ∼ 10r_s is as follows. In the region r < 10r_s, cooling is significantly stronger than heating, the gas is thermally unstable and high-density cold clumps form around the equatorial plane. In Figure 2, we plot the density at the equatorial plane and the θ angle averaged (or volume weighted) density for models NS1 and NS2. For model NS1, in the region r < 10r_s, the density at equatorial plane is 10 times higher than the θ angle averaged density. The presence of high-density cold clumps at equatorial plane makes the high contrast between density at equatorial plane and the θ angle averaged density. For a comparison, in model NS2, no cold clumps form at equatorial plane. Therefore, in model NS2, the density at equatorial plane is just 2 times higher than the θ angle averaged density. In model NS1, the presence of high-density cold clumps at the equatorial plane in the region r < 10r_s makes the radiation in this region significantly stronger. This is the reason why there is a sharp jump of f_c at r ∼ 10r_s. We note that in Pringle et al. (1973), it is anticipated that when the accretion rate reaches a high value, cold clumps will form in the inner region close to the central object.

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For an accretion flow, the internal energy equation is ${de}/{dt}+{pdv}={dQ}={Q}_{\mathrm{vis}}-{Q}_{\mathrm{cool}}$. In this equation, e is gas internal energy per volume, –pdv is compression work heating rate, Q_cool is the radiative cooling rate. In the accretion flow around a NS, ${Q}_{\mathrm{cool}}={Q}^{-}+{Sc}$. In the accretion flow around a BH, Q_cool = Q⁻. We can rewrite this equation as ${de}/{dt}={dQ}-{pdv}={Q}_{\mathrm{vis}}-{pdv}-{Q}_{\mathrm{cool}}$. In this equation, the term dQ (=Q_vis – Q_cool) can be positive, zero, or negative. It is not violating energy conservation law that if we have dQ to be negative. If the sum of the viscous heating and compression work heating terms (Q_vis – pdv) is smaller than radiative cooling term, it is also not violating the energy conservation law. In this case it means that with time evolution, the internal energy of gas will decrease. For the classic HAF model (e.g., Narayan & Yi 1994), accretion rate is extremely low, the term dQ is positive. With the increase of accretion rate, dQ can become negative. If dQ is negative and dQ – pdv is positive, radiative cooling rate is smaller than the sum of viscous heating rate and compression work heating rate, the accretion flow is luminous HAF" (LHAF; Yuan 2001, see also Yuan & Narayan 2014). In this case, the accretion flow can still be hot. The word "luminous" means that the luminosity of LHAF is much higher than that of the classic HAF. In black hole X-ray binaries, the hard state often reaches luminosities ∼10% of Eddington luminosity during hard-to-soft transition. Yuan & Zdziarski (2004) suggested that such luminous hard state systems, and also some Seyfert galaxies, may be explained by the LHAF model. This has been confirmed in detailed modeling of XTE J1550–564 (Yuan et al. 2007), where the X-ray spectrum is naturally explained by the LHAF model. If the accretion rate increases to a even higher value, the term dQ – pdv can be negative, in this case, HAF solution will not exist. For model NS2, we average for the whole accretion flow and find that the ratio of cooling rate to heating rate is 0.6. For model NS1, this ratio is 0.84.

The temperature of ions is determined by the viscous heating rate and the Coulomb collision cooling rate (see Equation (3)). Because the Coulomb collision cooling rate ${q}^{\mathrm{ie}}\propto {n}^{2}\propto {\dot{M}}^{2}$, with the increase of accretion rate, Coulomb collision cooling rate increases faster than viscous heating rate. Therefore, the temperature of ions decreases with the increase of mass accretion rate (see the left panel of the second row in Figure 1). The temperature of electrons almost does not depend on mass accretion rate. In the radial self-similar solution of accretion flows around a NS, Qiao & Liu (2018) also find that temperature of electrons is not sensitive to mass accretion rate. The reason is that for electrons at any mass accretion rate, the coulomb collision heating energy can be very efficiently lost by the radiative cooling. The temperature of electrons obtained in this paper is 2–3 times higher than what is observed in a NS accretion flow (Burke et al. 2017). The reason may be as follows. The temperature of electrons depends on heating and cooling of electrons. There are two heating sources for electrons. The first one is the energy transfer from ions to electrons by Coulomb collisions, which has been appropriately included in our simulations. The second one is the directly heating by viscous stress. In other words, one portion of viscous heating energy can go directly into electrons. Because, the exact fraction of viscous heating energy that directly goes into electrons is still unclear and under debate (see Yuan & Narayan 2014 for a review), in this paper, following Narayan & Yi (1995), we assume that all the viscous hearting energy goes into ions. In a realistic NS accretion system, may be a portion of viscous heating energy directly goes into electrons. In this case, because electrons can very quickly radiate away energy they receive, we may expect a higher radiative efficiency and more energy is lost from the accretion system by radiation. Correspondingly, the temperature of electrons in the realistic NS accretion system should be lower.

The gas pressure gradient force (per unit mass) which drives outflow/wind is $-{\rm{\nabla }}p/\rho \propto T$. In models NS2 and NS3, the temperature of ions is lower (compared with that in model NS3), the outflow/wind (compared with inflow) is weaker than that in model NS3. In model NS3, the outflow rate exceeds the net accretion rate significantly outside 25r_s. In models NS2 and NS3, the outflow rate just roughly equals to the net rate outside 25r_s. Therefore, in models NS2 and NS3, the mass inflow rate outside 20r_s is almost a constant with radius.

In the top right panel of Figure 1, we plot the volume-weighted radial profiles of density:

$$\begin{eqnarray}&&\rho (r)=\displaystyle \frac{2\pi {r}^{2}{\int }_{0}^{\pi }\rho \sin \theta d\theta }{2\pi {r}^{2}{\int }_{0}^{\pi }\sin \theta d\theta }.\end{eqnarray} \tag{ 15 }$$

The three dotted lines are radial power-law function fits to the density profiles. In model NS3, strong outflow/wind can take away mass, therefore the density increases slower inwards. The radial profile of density can be described as ρ ∝ r^−0.55. The result is consistent with that found in nonradiative simulations of HAF around a BH (e.g., Stone et al. 1999). In models NS1 and NS2, the outflow is weak (compared with inflow). The density increases faster inwards. The radial profile of density is ρ ∝ r^−p. The values of p are 1.4 and 1.2 for models NS1 and NS2, respectively. For a HAF without wind, we will have ρ ∝ r^−1.5 (e.g., Narayan & Yi 1994). The radial density profiles in models NS1 and NS2 are quite similar as that in models without wind.

The electron scattering optical depth in vertical direction τ_es–θ is an important parameter for calculating the spectrum of a HAF (Burke et al. 2017). It is defined as follows:

$$\begin{eqnarray}&&{\tau }_{{es}-\theta }={\int }_{0}^{\pi }\rho \kappa {rd}\theta .\end{eqnarray} \tag{ 16 }$$

In the right panel of the second row of Figure 1, we plot the radial profiles of τ_es–θ. As expected, the value of τ_es–θ increases with the increase of mass accretion rate.

The other parameter that affecting the spectrum of a HAF is the Compton y parameter (Burke et al. 2017). It is defined as follows:

$$\begin{eqnarray}&&y=\displaystyle \frac{4{{kT}}_{e}}{{m}_{e}{c}^{2}}\max ({\tau }_{{es}-\theta },{\tau }_{{es}-\theta }^{2})\end{eqnarray} \tag{ 17 }$$

where k and m_e are Boltzmann constant and mass of electron, respectively. The temperature of electrons is almost independent of mass accretion rate (see the left panel in the second row of Figure 1). Therefore, the value of y increases with mass accretion rate (see the right panel in the second row of Figure 1), because τ_es–θ increases with mass accretion rate.

In Paper I, we assume that electrons and ions have same temperatures and perform simulations of HAF around NS. Also, in Paper I, the radiative cooling term Q⁻(bremsstrahlung, synchrotron, and their self-Comptonization) is neglected. In paper I, we find that the soft photons from the surface of a NS can cool the gas via Comptonization. We find that for the one-temperature HAF, outflows are completely absent when $\dot{m}\gt {10}^{-4}$. However, in the present paper, we find that only when $\dot{m}\gt {10}^{-3}$, can outflows be significantly weak (models NS1 and NS2). In model NS3, with $\dot{m}=1.6\times {10}^{-4}$, outflows are very strong. We define a critical accretion rate (${\dot{m}}_{c}$) as the accretion rate above which outflows are significantly weak compared with inflows. The value of ${\dot{m}}_{c}$ in the present two-temperature simulations is one order of magnitude higher than that in the one-temperature simulations in Paper I. The reason is as follows. In the one-temperature simulations (Paper I), the soft photons from the surface of a NS can very effectively cool the electrons via Comptonization when $\dot{m}\gt {10}^{-4}$. Because in Paper I, we assume that ions have same temperature as electrons, therefore, ions are also have very low temperature. In the two-temperature simulations, the soft photons from the surface of a NS can only cool the electrons. The temperature of ions can still be very high even temperature of electrons is low. For example, in Paper I, when $\dot{m}=1.2\times {10}^{-4}$, the gas (ions and electrons) temperature at the inner radial boundary is 5 × 10¹⁰ K. In model NS3 with $\dot{m}=1.6\times {10}^{-4}$ in the present paper, the ions temperature at the inner radial boundary is 5 × 10¹¹ K. The gas pressure gradient force in model NS3 in the present paper can be one order of magnitude higher than that in the model with $\dot{m}=1.2\times {10}^{-4}$ in Paper I. This is the reason why the critical accretion rate ${\dot{m}}_{c}$ in the present paper is significantly higher than that in the one-temperature simulations in Paper I.

In models BH1, BH2, and BH3, the central object is a BH. All of the energy stored in the accretion flow will eventually go into the BH horizon. The cooling term Sc = 0 in models BH1, BH2, and BH3.

In the black hole accretion flow, one portion of the viscous heating energy is stored in gas as entropy and advected into the BH horizon. The other portion is lost by radiation of electrons (the term Q⁻). In model BH3 with the lowest accretion rate, we find that more than 80% of the viscous heating energy is stored in gas and advected into the black hole horizon; radiative cooling is negligible. Therefore, in model BH3, the dynamical properties of the HAF are quite similar as those of a nonradiative HAF. As mentioned above, for the accretion flow around a NS, when the accretion flow has $\dot{m}\lt \sim {10}^{-4}$ (model NS3), the dynamical properties of the HAF are also quite similar as those of a nonradiative HAF due to the very low cooling rate. Therefore, we can conclude that when $\dot{m}$ is smaller than $\sim {10}^{-4}$, the dynamical properties of a HAF around a NS are quite similar as those of a HAF around a BH. For example, we find that in model BH3, the radial profile of the mass inflow rate outside 10r_s can be described as ${\dot{M}}_{\mathrm{in}}\propto {r}^{s}$, with s = 0.57. In model NS3, the value of s is also 0.57. In model BH3, the density profile can be described as $\rho \propto {r}^{-p}$, with p = 0.55. In model NS3, the value of p is also 0.55.

We now compare the properties of the HAF in model NS2 to those of the HAF in model BH2. The values of $\dot{m}$ at the inner boundary in these two models are very similar. The results are shown in Figure 3. From the bottom two panels of Figure 3, we can see that in model NS2, the dominant cooling process is the Comptonization cooling (Sc) by collisions of electrons and the soft photons from the surface of the NS. In model BH2, the term Sc = 0. However, the value of f_c is much larger in model BH2 than that in model NS2. The gas density in NS2 is much higher than that in model BH2, one may expect that the value of f_c in model NS2 should be higher than that in model BH2. However, the trend is opposite. The reason is as follows. The exact formula of viscous heating rate is expressed in Equation (3). The viscous heating rate is proportional to $\rho /M{\left(r/{r}_{s}\right)}^{-2.5}$, where M is mass of the central object. For simplicity, we assume that the cooling is bremsstrahlung radiation. The cooling rate is proportional to ${\rho }^{2}{T}_{e}^{0.5}$. We have, ${f}_{c}\propto \rho {T}_{e}^{0.5}{\left(r/{r}_{s}\right)}^{2.5}M$. At a fixed radius (in unit of Schwarzschild radius), the temperatures of electrons in models NS2 and BH2 are almost same. The density in model BH2 is roughly 5 times smaller than that in model NS2. However, the black hole mass in BH2 is roughly 7 times higher than the NS mass in model NS2. Therefore, we would expect a slightly higher value of f_c in model BH2 than that in model NS2. In reality, if we consider all the radiative cooing process (bremsstrahlung, synchrotron, and their Comptonization), the situation will be more complex. It is hard to expect that with a similar accretion rate (in unit of Eddington rate), in which model (BH or NS accretion flow), the value of f_c will be larger. In model BH2, inside ∼20r_s, f_c ∼ 0.5 and outside 20r_s, the value of f_c ∼ 1. Inside ∼20r_s, about 50% of the viscous heating energy is lost by radiative cooling (${Q}^{-}$). Outside 20r_s, all of the viscous heating energy is lost by radiative cooling (${Q}^{-}$). In model NS2, the term Sc is the dominant cooling term. In model NS2, inside ∼20r_s, f_N ∼ 0.5 and outside 20r_s, the value of f_N > 1. In model NS2, inside ∼20r_s, about 50% of the viscous heating energy is lost by the cooling term Sc. Outside 20r_s, all of the viscous heating energy is lost by the cooling term Sc. From the comparisons we can see that the fraction of viscous heating energy that lost by cooling in model NS2 is very similar as that in model BH2. In other words, the fraction of viscous heating energy that stored in gas as entropy in model NS2 is very similar as that in model BH2. The properties of the HAF in model NS2 are quite similar as those in model BH2. The absolute vale of density in model NS2 is higher than that in model BH2 (see the top right panel in Figure 3). The reason is as follows. The mass of the BH is much larger than that of the NS. The values of $\dot{m}$ in the two models are quite similar. Therefore, the absolute value of density in model NS2 is much higher than that in model BH2. However, the slope of the density profile in model NS2 is almost same as that of the density profile in model BH2. In both models of BH2 and NS2, outflow is weak and the radial profile of mass inflow rate is quite flat. The temperature of ions in model NS2 is almost same as that in model BH2. The temperature of electrons in model NS2 is just lightly lower than that in model BH2. The electron scattering optical depths (τ_es(r) and τ_es–θ) in both r and θ directions in model BH2 are also slightly larger than those in model NS2. The Compton y parameter in model BH2 is slightly higher.

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Figure 4 shows the comparisons of models NS1 and BH1. The values of $\dot{m}$ at the inner boundary in these two models are very similar. From the bottom panels we can see that inside 10r_s in both models, the value of f_c is significantly larger than 1. Therefore, in this region, radiative cooling (Q⁻) dominates the viscous heating. Also, in this region (r < 10r_s), the ratios of the cooling rate to viscous heating rate in the two models are very similar. Outside 10r_s, the ratios of cooling rate to the heating rate in the two models are also very similar. For example, in model BH1, at 100r_s, the ratio of cooling rate to viscous heating rate f_c ∼ 1.5; in model NS1, at the same radius, this ratio is f_c + f_N ∼ 1.8. The properties of the accretion flow in model NS1 are also quite similar as those in model BH1. The Comptonized spectrum is mainly determined by the Compton y parameter, and is slightly affected by the temperature of the seed photons. The Compton y parameter in model NS1 is almost same as that in model BH1. Therefore, we can expect that the shape of the Comptonized spectrum (or hard X-ray spectrum) in model NS1 should be almost same as that in model BH1. However, in the NS accretion flow spectrum, there is a peak in the soft X-ray band due to the presence of soft photons from the surface of the NS. In the BH accretion flow spectrum, there is no such peak in the soft X-ray band. In this sense, the spectrum of a NS system is different from that of a BH system.

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From Figures 3 and 4, we can see that the BH models and NS models have almost same electron temperature. Usually, it is expected that electron temperature of a NS accretion flow is much lower than that of a BH accretion flow. The reason is that, the soft photons from the surface of a NS can effectively cool the accretion flow. For a BH accretion flow, there is no such soft photons. Whether is this expectation valid depends on accretion rate. For low accretion rate systems, this expectation is valid. The reason is as follows. For low accretion rate BH system (e.g., model BH3), the radiation of electrons (bremsstrahlung, synchrotron, and their Comptonization) is not so important, the electrons have higher temperature. For a NS system with a similar accretion rate (e.g., model NS3), the radiation of electrons (bremsstrahlung, synchrotron, and their Comptonization) is also not so important. However, the soft photons from the surface of the NS can effectively cool electrons and make temperature of electrons lower. We find that the electron temperature in model NS3 is two times lower than that in model BH3. For high accretion rate system, the expectation that NS system has a much lower electron temperature than BH system is not valid. The reason is as follows. In the high accretion rate BH system (e.g., models BH1 and BH2), the radiation of electrons (bremsstrahlung, synchrotron, and their Comptonization) is very strong and the radiative efficiency is very high. In a NS system with a similar accretion rate, the radiation of electrons (bremsstrahlung, synchrotron, and their Comptonization) is also very strong and the presence of soft photons from the surface of a NS cannot significantly affect the radiative efficiency of the accretion flow. This is the reason why for the high accretion rate NS and BH systems (models NS1, NS2, BH1, and BH2), the temperatures of electrons are quite similar.