Observability of Single Neutron Stars at SRG/eROSITA

Abstract

A four-year sky survey with the use of the eROSITA telescope on board the Spektr-RG observatory with focusing optics will provide the best coverage in the soft (0.5–2 keV) and standard (2–10 keV) X-ray ranges, both in terms of sensitivity and angular resolution. We have investigated the possibility of detecting various types of single neutron stars using eROSITA. Among the already known objects, eROSITA will be able to detect more than 160 pulsars, 21 magnetars, 7 central compact objects, all seven sources of the Magnificent Seven, and two other X-ray single neutron stars during the four-year mission. In addition, eROSITA is expected to be able to detect accreting single neutron stars, as well as discover new cooling neutron stars and magnetars.

Appendix

To proceed from Eq. (16) to Eq. (17), we use the relationship between the redshift $z$ and the scale factor a:

$$a = 1{\text{/}}(1 + z).$$

Differentiating the scale factor over time, we obtain

$$\frac{{da}}{{dt}} = - \frac{1}{{{{{(1 + z)}}^{2}}}}\frac{{dz}}{{dt}}.$$

Therefore,

$$\frac{{dz}}{{dt}} = - {{(1 + z)}^{2}}\frac{{da}}{{dt}}.$$

By definition, the Hubble constant $H$ is

$$H = \dot {a}{\text{/}}a.$$

Let us substitute this expression into the formula for $\dot {z}$:

$$\frac{{dz}}{{dt}} = - {{(1 + z)}^{2}}\frac{{da}}{{dt}} = - {{(1 + z)}^{2}}H(z)a = - H(z)(1 + z).$$

The Hubble constant depends on time and can be written using the well-known expression:

$$\begin{gathered} H(z) = {{H}_{0}}[{{\Omega }_{r}}{{(1 + z)}^{4}} \\ \, + {{\Omega }_{m}}{{(1 + z)}^{3}} + {{\Omega }_{c}}{{(1 + z)}^{2}} + {{\Omega }_{\Lambda }}{{]}^{{1/2}}}. \\ \end{gathered} $$

Here, ${{H}_{0}}$ is the Hubble constant at the present point of time, and in square brackets are the terms associated with the contribution of various components to the total density: radiation (${{\Omega }_{r}}$), matter (${{\Omega }_{m}}$), curvature (${{\Omega }_{c}}$), and cosmological constant (${{\Omega }_{\Lambda }}$)

$$\begin{gathered} \frac{{dz}}{{dt}} = - H(z)(1 + z) - {{H}_{0}}(1 + z)[{{\Omega }_{r}}{{(1 + z)}^{4}} \\ \, + {{\Omega }_{m}}{{(1 + z)}^{3}} + {{\Omega }_{c}}{{(1 + z)}^{2}} + {{\Omega }_{\Lambda }}{{]}^{{1/2}}}. \\ \end{gathered} $$

Now, we obtain

$$\frac{{dz}}{{dt}} = - {{H}_{0}}(1 + z){{[{{\Omega }_{m}}{{(1 + z)}^{3}} + {{\Omega }_{\Lambda }}]}^{{1/2}}},$$

where it is taken that ${{\Omega }_{r}} = {{\Omega }_{c}} = 0$. Now, we express the local time interval in terms of the redshift interval:

$$dt = - dz\frac{1}{{{{H}_{0}}(1 + z){{{[{{\Omega }_{m}}{{{(1 + z)}}^{3}} + {{\Omega }_{\Lambda }}]}}^{{1/2}}}}}.$$

Substituting this into the column density formula, we obtain

$${{N}_{{\text{H}}}} = - \int\limits_z^0 \,{{n}_{0}}{{a}^{{ - 3}}}\frac{c}{{{{H}_{0}}}}{{[{{\Omega }_{m}}{{(1 + z)}^{3}} + {{\Omega }_{\Lambda }}]}^{{ - 1/2}}}{{(1 + z)}^{{ - 1}}}dz.$$

Here, ${{n}_{0}}$ is the average density of the intergalactic medium at the present time. Note that the absorbing matter is distributed unevenly (see the reviews [82, 83]). Now, we replace the scale factor in the equation with $a = 1{\text{/}}(1 + z)$ and obtain the final formula for the dependence of the column density in the intergalactic medium on the redshift $z$:

$${{N}_{{\text{H}}}} = - \frac{{{{n}_{0}}c}}{{{{H}_{0}}}}\int\limits_z^0 \,\frac{{{{{(1 + z)}}^{2}}}}{{{{{[{{\Omega }_{m}}{{{(1 + z)}}^{3}} + {{\Omega }_{\Lambda }}]}}^{{1/2}}}}}dz.$$

(19)