Theoretical background and results

In this section we consider the pulsars and estimate the corresponding physical parameters of artificial ring-like constructions surrounding the rotating neutron stars.

Generally speaking, any rotating neutron star, characterized by the slow down rate ${\dot P} \equiv {\rm d}P/{\rm d}t \gt 0$ , where P is the rotation period, loses energy with the following power (called the slowdown luminosity, L _sd) $\dot W = I\Omega \vert \dot \Omega \vert $ . Here by $I = 2MR_{\rm \ast} ^2 /5$ we denote the moment of inertia of the neutron star, $M \approx 1.5 \times {M_{\rm \odot}} $ and ${M_{\rm \odot}} \approx 2 \times {10^{33}}$ g are the pulsar's mass and the solar mass, respectively, R _* ≈ 10⁶ cm is the neutron star's radius, Ω ≡ 2π/P is its angular velocity and ${\dot \Omega} \equiv {\rm d\Omega} /{\rm d}t = - 2{\rm \pi} {\dot P} /{P^2}$ . The slowdown luminosity for the relatively slowly spinning pulsars is of the order of

(1) $$\eqalign{ {L_{\rm sd}} \approx & 4.7 \times {10^{31}} \times {P^{ - 3}} \times \left(\displaystyle{{\dot P} \over {{{10}^{ - 15}}{\rm ss^{ - 1}}}}\right) \cr & \times \left(\displaystyle{M \over {1.5{M_ \odot }}}\right){\rm ergs}\;\;{{\rm s}^{ - 1}},} $$

where the parameters are normalized on their typical values. As it is clear from equation (1), the energy budget is very high forcing a supercivilization construct a ‘ring’ close to the host object. On the other hand, these sources exist during the time scale $P/{\dot P} $ , which is long enough to colonize the star.

In the framework of the standard definition, a HZ is a region of space with favourable conditions for life based on complex carbon compounds and on availability of fluid water, etc. (Hanslmeier Reference Hanslmeier2009). This means that the surface of the ‘ring’ must be irradiated by the same flux as the surface of the Earth (henceforth – the flux method). Therefore, the mean radius of the HZ, is defined as ${R_{{\rm HZ}}} = {({\rm \kappa} {L_{{\rm sd}}}/{L_{\rm \odot}} )^{1/2}}\,{\rm AU}$ , where ${L_{\rm \odot}} \approx 3.83 \times {10^{33}}\,{\rm ergs}\,{{\rm s}^{ - 1}}$ is the solar bolometric luminosity, is expressed as follows

(2) $$\eqalign{{R_{\rm HZ}} \approx &\;\; 3.5 \times {10^{ - 2}} \times {P^{ - 3/2}} \times {\left(\displaystyle{{\rm \kappa} \over {0.1}} \right)^{1/2}} \cr & \quad \times {\left(\displaystyle{{\dot P} \over {{{10}^{ - 15}}{\rm ss}^{-1}}} \right)^{1/2}} \times {\left(\displaystyle{M \over {{1.5}{M_{\rm \odot}}}} \right)^{1/2}} {\rm AU},} $$

where we have taken into account that the bolometric luminosity, L, of the pulsar is less than the slowdown luminosity and is expressed as κL _sd, κ < 1. As we see, the radius of the HZ for 1 s pulsars is very small compared with the radius of the typical Dyson sphere – 1 AU. The best option for the supercivilization could be to find a pulsar with an angle between the magnetic moment (direction of one of the channels) and the axis of rotation close to 90°, because in this case an artificial ‘ring’ has to be constructed in the equatorial plane. In case of relatively small inclination angles, there should be two rings, each of them shifted from the equatorial plane, although, in this case it is unclear how do the ‘rings’ keep staying in their planes, therefore, we focus only on pulsars with α ≈ 90°.

According to the standard model of pulsars it is believed that the radio emission is generated by means of the curvature radiation defining the opening angle of the emission channel (Ruderman & Sutherland Reference Ruderman and Sutherland1975)

(3) $$\eqalign{{\rm \beta} \approx &\; {32^{\circ}} \times {P^{-13/21}} \times {\left( \displaystyle{{\rm \rho} \over {{10}^{6}\,{\rm cm}}} \right)^{2/21}} \cr & \quad \times {\left(\displaystyle{{\dot P} \over {{10}^{-15}{\rm ss}^{-1}}} \right)^{1/14}} \times {\left(\displaystyle{{\rm \omega} \over {1.0 \times {10}^{10}\,{\rm Hz}}}\right)^{-1/3}},} $$

where the radius of curvature of magnetic field lines – ρ and the cyclic frequency of radio waves – ω, are normalized on their typical values (Ruderman & Sutherland Reference Ruderman and Sutherland1975). One can straightforwardly check that the artificial ‘ring’ with equal radiuses of the spherical segment bases should have height in the following interval (1−2)R _HZsin(β)≈(0.006−0.06)AU.

One of the significant parameters to search the ‘rings’ is their temperature, which can be estimated quite straightforwardly. In particular, if the aim of the super civilization is to consume the total energy radiated by the host pulsar, then albedo of the material the ‘ring’ has to be made of must be close to zero. For simplicity we consider α = 90°, then it is clear that the inner surface of the ‘ring’, irradiated by the pulsar, every single second absorbs energy of the order of L. On the other hand, energy balance requires that the ‘ring’ must emit the same amount of energy in the same interval of time, L = A _efσT ⁴, leading to the following expression

(4) $$T = {\left( {\displaystyle{L \over {{A_{_{{\rm ef}}}} {\rm \sigma}}}} \right)^{1/4}},$$

where σ ≈ 5.67 × 10⁻⁵ erg/(cm² K⁴) is the Stefan–Boltzmann constant, ${A_{_{{\rm ef}}}} = 8{\rm \pi} R_{_{{\rm HZ}}} ^{\rm 2} \sin \left( {{\rm \alpha} /2} \right)$ is the effective area of the spherical segment taking into account inner and outer surfaces and T is the average temperature of the ‘ring’.

After applying equations (1)–(3), one can obtain T. In Fig. 2 (top panel) we plot the behaviour of temperature versus the period of rotation of a pulsar for different values of the slow down rate ${\dot P}$ : ${\dot P}{ \;= 10^{ - 15}}\;{\rm s}{{\rm s}^{ - 1}}$ (solid line); ${\dot P}\; { = 10^{ - 14}}\;{\rm s}{{\rm s}^{ - 1}}$ (dashed line); ${\dot P} = 2 \times {10^{ - 14}}\;{\rm s}{{\rm s}^{ - 1}}$ (dotted-dashed line). As we see from the graph, for relatively slowly spinning pulsars the typical values of the effective temperature of the artificial construction are in the following interval ~(400–500) K. The corresponding radius of the HZ varies from ~10⁻² to ~10⁻¹ AU (see the bottom panel).

Fig. 2. On the top panel, in the framework of the flux method, we plot the dependence of T on P for three different values of ${\dot P}$ : ${\dot P} = 10^{ - 15}\;{\rm s}{{\rm s}^{ - 1}}$ (solid line); ${\dot P} = 10^{ - 14}\;{\rm s}{{\rm s}^{ - 1}}$ (dashed line); ${\dot P} = 2 \times {10^{ - 14}}\;{\rm s}{{\rm s}^{ - 1}}$ (dotted-dashed line). As it is clear from the graph, for typical values of relatively slowly spinning pulsars the effective temperature of the artificial construction ranges from ~400 to ~500 K. On the bottom panel, in the framework of the temperature method, we show the dependence of R _HZ on T for the same values of $\dot P$ . As we see the distance to the HZ ranges from 2 × 10⁻⁴ to 1.3 × 10⁻³ AU.

Generally speaking, the distance to the HZ is not strictly defined, because our knowledge about life is very limited by the conditions we know on Earth. But even in the framework of life on Earth the radius of the HZ might be defined in a different way, assuming a distance enabling the effective temperature in the interval: (273–373) K where water can be in a liquid state (henceforth – the temperature method)Footnote ¹ . From equation (4) one can show that

(5) $${R_{{\rm HZ}}} = {\left( {\displaystyle{L \over {8{\rm \pi \sigma} \sin \left[ {{\rm \beta} /2} \right]{T^4}}}} \right)^{1/2}}.$$

In Fig. 2 (bottom panel) we plot the graphs of R _HZ versus T for the same values of ${\dot P}$ . It is evident that in this case the distance to the HZ ranges from 2 × 10⁻⁴ to 1.3 × 10⁻² AU.

Another possibility for the supercivilization might be the ‘colonization’ of millisecond pulsars since these objects reveal extremely high values of luminosity. In particular, for the pulsar with P = 0.01 s and ${\dot P}{ \;= 10^{ - 13}}\;{\rm s}{{\rm s}^{ - 1}}$ the slowdown luminosity is of the order of 9.5 × 10³⁸ ergs s⁻¹  and by taking into account κ ≈ 0.01 (being quite common for millisecond pulsars), one can show that the bolometric luminosity

(6) $$\eqalign{L \approx\; & 9.5 \times {10^{36}} \times {\left(\displaystyle{P \over {0.01{\rm s}}} \right)^{-3}} \times {\left(\displaystyle{{\rm \kappa} \over {0.01}} \right)} \cr & \quad \times {\left( \displaystyle{{\dot P} \over {{10}^{-13}{\rm ss}^{-1}}}\right)} \times {\left(\displaystyle{M \over {{1.5}M_{\rm \odot}}}\right)}{\rm ergs}\;{\rm s}^{-1},} $$

is by several orders of magnitude higher than for relatively slowly rotating pulsars. Here the physical quantities are normalized on typical values of rapidly rotating neutron stars. It is clear that all rapidly spinning pulsars have extremely high energy output in the form of electromagnetic waves. As a result, a corresponding distance from the central object to the HZ must be much bigger than for slowly rotating pulsars.

In order to estimate the height of the ‘ring’ one has to define the opening angle of a radiation cone for millisecond pulsars radiating in the X-ray spectrum. According to the classical paper (Machabeli & Usov Reference Machabeli and Usov1979), the X-ray emission of pulsars has the synchrotron origin, maintained by the quasi-linear diffusion. As it was shown by Machabeli & Usov (Reference Machabeli and Usov1979) the corresponding angle of the radiation cone is given by

(7) $${\rm \beta} \approx {8^ {\circ}} \times {\left( {\displaystyle{{0.01\,{\rm s}} \over P}} \right)^{1/2}} \times {\left( {\displaystyle{{{R_{{\rm st}}}} \over {{{10}^6}\,{\rm cm}}}} \right)^{1/2}},$$

which automatically gives an interval of height of the artificial construction: (0.02–0.15) AU.

After combining Equations (4) and (7), like the previous case of slowly rotating neutron stars, one can estimate the effective temperature for millisecond pulsars. In particular, in Fig. 3 (top panel) in the framework of the flux method, we show the behaviour of temperature of the ‘ring’ as a function of P and as we see, for the following interval of rotation periods: (0.01–0.05) s the temperature ranges from ~540 K to approximately 660 K. The size of the HZ ranges from 30 to 350 AU. Contrary to this, by applying the temperature method, in Fig. 3 (bottom panel) we show the behaviour of R _HZ versus T for three different values of ${\dot P}$ : ${\dot P}{ \;= 10^{ - 13}}\;{\rm s}{{\rm s}^{ - 1}}$ (solid line); ${\dot P} = 2 \times {10^{ - 13}}\;{\rm s}{{\rm s}^{ - 1}}$ (dashed line); $ {\dot P} = 4 \times {10^{ - 13}}\;{\rm s}{{\rm s}^{ - 1}}$ (dotted-dashed line). From the figure it is clear that the distance to the HZ ranges from 10 to 30 AU.

Fig. 3. On the top panel we present the behaviour of T versus P in the framework of the flux method. It is evident that for rapidly spinning pulsars the effective temperature of the ‘ring’ is in the following interval: (540–660) K. On the bottom panel, in the framework of the temperature method, we show the dependence of R _HZ on T for three different values of ${\dot P}$ : ${\dot P}{ \;= 10^{ - 13}}\;{\rm s}{{\rm s}^{ - 1}}$ (solid line); ${\dot P} = 2 \times {10^{ - 13}}\;{\rm s}{{\rm s}^{ - 1}}$ (dashed line); ${\dot P} = 4 \times {10^{ - 13}}\;{\rm s}{{\rm s}^{ - 1}}$ (dashed-dotted line). As we see the distance to the HZ ranges from 10 to 30 AU.

To understand how realistic are these constructions it is interesting to estimate their masses. Let us assume that the material the rings are made of has the density of the order of the Earth, ρ ~ 5.5 g cm⁻³. Then it is straightforward to show that the expression of mass is given by

(8) $${M_{{\rm ring}}} \approx 4{\rm \pi \rho} R_{_{{\rm HZ}}} ^2 \Delta R\sin \left( {{\rm \beta} /2} \right),$$

where ΔR is the average thickness of the ring. If we consider slowly rotating pulsars, by assuming ΔR ~ 10 m and ${R_{_{{\rm HZ}}}} {\rm \sim} {10^{ - 3}}\,{\rm AU}$ (see Fig. 2), one can show that M _ring ≈ 4 × 10²⁴ g, that is three orders of magnitude less than the mass of the Earth. Therefore, in the planetary system the material could be quite enough to construct the ring-like structure around 1 s pulsars.

The similar analysis for millisecond pulsars, shows that the mass of the ring, 4 × 10³² kg, exceeds the total mass of all planets, planetoids, asteroids, comets, centaurs and interplanetary dust in the solar system by three orders of magnitude. This means that nearby regions of millisecond pulsars hardly can be considered as attractive sites for colonization.

Another issue one has to address is the force acting on the structure by means of the radiation. By assuming that half of the total energy is radiated in each of the radiation cones, the corresponding force can be estimated as follows

(9) $${F_{{\rm rad}}} \approx \displaystyle{L \over {2c}}.$$

On the other hand, the stress forces in the ring, caused by gravitational forces should be of the order of the gravitational force acting on the mentioned area of the ring. This force is given by

(10) $${F_{\rm g}} \approx \displaystyle{{GM{\rm \lambda} A} \over {R_{_{{\rm HZ}}} ^2}}, $$

where λ is the mass area density of the ring and $A = 2{\rm \pi} R_{_{{\rm HZ}}} ^2 (1 - \cos \left( {{\rm \beta} /2} \right))$ is the area of the ring, irradiated by the pulsar. It is clear that for maintaining stability of the structure the radiation force should be small compared with the gravitational force. By imposing this condition equations (9) and (10) lead to ${\rm \lambda}\ {\rm \gg}\ L/\left( {4{\rm \pi} GMc(1 - \cos \left( {{\rm \beta} /2} \right))} \right)$ , which for β = 32° reduces to

(11) $${\rm \lambda} \,{\rm \gg}\, 3.4 \times {10^{ - 6}}\displaystyle{{{L_{31}}} \over {{M_{1.5}}}}{\rm g}\;{\rm c}{{\rm m}^{ - 2}},$$

where L ₃₁ ≡ 10³¹ ergs s⁻¹ and ${M_{1.5}} \equiv M/(1.5{M_{\rm \odot}} )$ . As we see from the aforementioned estimate, for realistic scenarios the emission cannot significantly perturb the Dyson construction.

Generally speaking, rotation of pulsars drives high energy outflows of plasma particles around the star (Gudavadze et al. Reference Gudavadze, Osmanov and Rogava2015) and the influence of these particles deserves to be considered as well. The similar result can be obtained by considering the interaction of the pulsar wind with the ring. In particular, in equation (11) instead of the radiation term, L/2, one should apply the pulsar wind's kinematic luminosity, that cannot exceed the slowdown luminosity. After considering the maximum possible kinematic wind luminosity, L _sd, the critical surface density will be of the same order of magnitude. Therefore, the Dyson structure can survive such an extreme pulsar environment.

On the other hand, it is clear that gravitationally such a system might not be stable. In particular, McInnes (Reference McInnes2003) has considered a point mass and a thin solid ring having initially coincident centers of masses, thus being in the equilibrium state. Although, it has been shown that this equilibrium configuration is stable due to the perturbations normal to the plane of the ring and is unstable for perturbations within the mentioned plane.

In Fig. 4 we schematically show the in-plane displaced ring with respect to the pulsar. The gravitational force between the pulsar and the ring element with mass dm

(12) $$df = G\displaystyle{{Mdm} \over {{{r^{\prime}}^2}}},$$

projected on the r line and integrated over the hole ring (McInnes Reference McInnes2003)

(13) $${\,f_r}({\rm \zeta} ) = G\displaystyle{{M{M_{{\rm ring}}}} \over {2{\rm \pi} {R^2}}}\int_0^{2{\rm \pi}} \displaystyle{{{\rm \zeta} + \cos {\rm \phi}} \over {{{\left( {1 + 2{\rm \zeta} \cos {\rm \phi} + {{\rm \zeta} ^2}} \right)}^{3/2}}}}d{\rm \phi}, $$

for small values of ζ ≡ r/R, leads to the equation describing the in-plane dynamics (McInnes Reference McInnes2003)

(14) $$\displaystyle{{{d^2}{\rm \zeta}} \over {d{t^2}}} - \displaystyle{1 \over {{{\rm \tau} ^2}}}{\rm \zeta} = 0,$$

where ${\rm \tau} \equiv \sqrt {2{R^3}/(GM)} $ is the timescale of the process. This equation has the following solution

(15) $${\rm \zeta} (t) = {{\rm \zeta} _0}{{\rm e}^{t/{\rm \tau}}}, $$

indicating that the equilibrium configuration is unstable against in-plane perturbations. Here, ζ₀ is the initial nondimensional perturbation. By means of this process the ring gains the kinetic energy with the following power

(16) $$W = {M_{{\rm ring}}}{R^2}\displaystyle{{d{\rm \zeta}} \over {dt}}\displaystyle{{{d^2}{\rm \zeta}} \over {d{t^2}}},$$

therefore, in order to restore the equilibrium state of the ring, one should utilize the same amount of energy from the pulsar corresponding to the instability timescale. On the other hand, it is evident that such constructions may have sense only if the power needed to maintain stability is small compared with the bolometric luminosity of the pulsar. By taking into account this fact, one can straightforwardly show that the initial declination from the equilibrium position must satisfy the condition

(17) $${{\rm \zeta} _0}\ {\rm \ll}\ \displaystyle{{0.37} \over R}{\left( {\displaystyle{{{L_{\rm b}}} \over {{M_{{\rm ring}}}}}} \right)^{1/2}}{\left( {\displaystyle{{2{R^3}} \over {GM}}} \right)^{3/4}},$$

which for the parameters presented in Fig. 3 leads to ${{\rm \zeta} _0}\ {\rm \ll}\ ({10^{ - 5}} - {10^{ - 4}})$ . It is worth noting that such a precision of measurement of distance for supercivilization cannot be a problem. For instance, in the Lunar laser ranging experimentFootnote ² the distance is measured with the precision of the order of 10⁻¹⁰.

Fig. 4. Here we show the in-plane displaced ring with respect to the pulsar, denoted by M.

In the vicinity of pulsars radiation protection could be one of the important challenges facing the super advanced civilization. In particular, the radiation intensity

(18) $$I = \displaystyle{{{L_{\rm b}}} \over {4{\rm \pi} {R^2}\left( {1 - \cos \left( {{\rm \beta} /2} \right)} \right)}},$$

for the shortest ring radius 2 × 10⁻⁴ AU is of the order of 10¹² erg s⁻¹ cm⁻² and therefore, it should be of great importance to protect the civilization from high energy gamma rays. For this purpose one can use special shields made of certain material, efficiently absorbing the radiation, which in turn, might significantly reduce the corresponding intensity. If one uses half-value layer, HVL, which is the thickness of the material at which the intensity is reduced by one half we can estimate if the thickness of the ring is enough to make a strong protection from extremely high level of emission. If as an example we examine concrete with HVL = 6.1, one can show that a layer of thickness 2.5 m reduces the intensity by 10¹² orders of magnitude, being even more than enough for radiation protection.