Main consideration

In this section we consider a theoretical model of DSs with reduced radii, estimate corresponding temperatures and the consequent emission spectra and study the stability of the megastructure.

Unlike the case examined by Dyson (Reference Dyson1960), where the DSs were located in the star's HZ in this section, we consider a sphere with higher surface temperature, T. This could be quite reasonable, because smaller megastructure requires less material. By assuming the black body radiation one can show that the radius of the DS is given by

(1)$$ \eqalign{R & = \left(\displaystyle {L} \over {4 \pi \sigma T^4}\right)^{1/2} \cr & \quad \approx 2.14 \times 10^{12} \times \left(\displaystyle{L} \over {L_{\odot}}\right)^{1/2} \times \left(\displaystyle{1000\, {\rm K} \over {T}}\right)^2\, {\rm cm},} $$

where σ ≈ 5.67 × 10⁻⁵ erg  (cm² K⁴)⁻¹ is the Stefan-Boltzmann's constant and $L_{\odot }\approx 3.83\times 10^{33}$ erg⁻¹ is the solar luminosity. As it is clear from the above estimate the radius is almost one orders of magnitude less than one astronomical unit, a typical size of the DS in the HZ. Here we used two assumptions: (I) the surface temperature is less than the melting point of a material the DS is made of and (II) the super civilization is capable of constructing an efficient cooling system inside the shell. Generally speaking, it is clear that a Level-II civilization might use metamaterials with high melting temperatures. In particular, by considering graphene it is straightforward to show that to maintain a cold region with a temperature of the order of 300 K the corresponding heat flux power (normalized on the solar luminosity) is given as

(2)$$ \eqalign{P_{{\rm c}} & \approx {\displaystyle{\kappa S} \over {L_{\odot}}} {\displaystyle{\Delta T} \over {h}} \approx 2.4 \times 10^{-6} \cr & \quad \times {\displaystyle{\Delta T/h} \over {70\,{\rm K}\,{\rm cm}}^{-1}} \times {\displaystyle{\kappa} \over {2.5 \times 10^8\,{\rm erg}}\,({\rm cm\,K})^{-1}} \times {\displaystyle{S} \over {S_{{\rm E}}}}{\rm erg}\,{\rm s}^{-1},}$$

where the temperature gradient is calculated for ΔT = (1000 − 300)K = 700 K, h = 100 cm and κ = 2.5 × 10⁸ ergs cm⁻¹ K⁻¹ is the typical value of the thermal conductivity of graphene (Cai et al. Reference Cai2010) and S – the area occupied by the extraterrestrials is normalized by the total surface area of Earth S _E ≈ 5.1 × 10⁸ km². If one assumes that the coefficient of performance (COP) for a cooling system is of the order of 5 (typical values of modern refrigerators), the engine, to compensate the aforementioned flux to the cold area must process the energy from the cold reservoir to the hot one

(3)$$ \eqalign{P_{{\rm e}} & = {\displaystyle{P_{{\rm c}} \over {{\rm COP}}}} \approx 4.7 \times 10^{-7} \times {\displaystyle{5} \over {{\rm COP}}} \cr & \quad \times {\displaystyle{\Delta T/h} \over {70\,{\rm K}\,{\rm cm}}^{-1}} \times {\displaystyle{\kappa} \over {2.5 \times 10^8\,{\rm erg}\,({\rm cmK})^{-1}}} \times {\displaystyle{S} \over {S_{{\rm E}}}} {\rm erg}\,{\rm s}^{-1}.} $$

As we see from this estimate, only a tiny fraction of the total luminosity is required to maintain a HZ inside the DS if one uses metamaterials (graphene), which Type-I civilization can produce. Therefore, there is no question if the Level-II extraterrestrials can produce such materials. One has to note that although temperature dependence for graphene is not studied for a wide range of temperatures, a preliminary study shows that for high temperatures κ should decrease (Pop et al. Reference Pop, Varshney and Roy2012). Therefore, the corresponding value of P _e might be even less. It is worth noting that if the primary purpose to construct the DS is computation, the aforementioned estimates should be changed (Sandberg Reference Sandberg1999).

An important issue we would like to address is the stability problem of the DSs. In an ideal case the star has to be located in the centre of the sphere, but it is clear that a physical system can remain in the equilibrium state only in the absence of disturbances. In Fig. 1 we schematically show the position of the star (point S) and the DS with the centre O. By assuming that the DS is shifted by x with respect to the equilibrium position we intend to study its dynamics of motion. It is obvious that according to the Gauss' law the gravitational interaction of the star and the spherical shell is zero. On the other hand, stars emit enormous energy in the form of electromagnetic radiation, which will inevitably act on the internal surface of the DS and consequently, the part of the surface which is closer to the star will experience higher pressure than the opposite side of the sphere. Therefore the restoring force will appear leading to the periodic motion of the megastructure.

By assuming the isotropic radiation, the corresponding pressure in the direction of emission (See the arrow in Fig. 1) is given by

(4)$$ P = {\displaystyle{L} \over {4\pi r^{2}c}}, $$

where

(5)$$ r = (x^2 + R^2 - 2xR \cos \varphi)^{1/2} $$

is the distance from the centre of the radiation source.

Fig. 1. Here we schematically show the DS with the centre O and a star S inside it.

If one assumes that the inner surface of the DS completely absorbs the incident radiation, the corresponding force acting on the differential surface area dA is given by

(6)$${\rm d}F = {\displaystyle{L} \over {4 \pi r^2c}} {\rm d}A \cos \gamma, $$

where γ is the angle $\angle OBS $. On the other hand, from the symmetry it is certain that the sphere will move along the x axis (coincident with the line OS). Therefore, the dynamics are defined by the component of the force along the mentioned direction. Then, by considering a ring with radius 2πRsin φ it is straightforward to show that

(7)$$dF_{x} = {\displaystyle{L} \over {4 \pi r^2c}}2\pi R^2 \sin \varphi \cos \gamma \cos \theta d \varphi. $$

After taking into account the relations

(8)$$ \theta = \varphi + \gamma, $$

and

(9)$$\cos \gamma = {\displaystyle{R-x\cos\varphi} \over {(x^2 + R^2 - 2xR \cos \varphi)^{1/2}}} $$

the x component of the total radiation force acting on the DS

(10)$$ F_{x} = {\displaystyle {LR^2} \over {2c}} \int_0^{\pi} {\displaystyle{\sin \varphi(R\cos \varphi - x)(R - x \cos \varphi)} \over {(x^2 + R^2 - 2xR \cos \varphi)^{2}}} d\varphi, $$

for small oscillations x/R ≪ 1 reduces to

(11)$$ F_{{x}} = -{\displaystyle{4L} \over {3c}} {\displaystyle{x} \over {R}}. $$

The minus sign clearly indicates that the force has a restoring character and consequently the dynamics of the DS is described by the following differential equation

(12)$$ {\displaystyle{dx^2} \over {dt^2}} + {\displaystyle{4L} \over {3cRM}} x = 0, $$

where M = 4πρR ²ΔR is the mass of the megastructure, ΔR is its thickness and ρ is the density of a material the DS is made of. By combining this expression with the aforementioned equation one can show that the period of oscillation is given by

(13)$$\eqalign{P_{{\rm DS}} & = 2\pi \left({\displaystyle{3\pi\rho cR^3\Delta R} \over {L}}\right)^{1/2} \approx 33.8 \times \left({\displaystyle{L} \over {L_{\odot}}}\right)^{1/4} \cr & \quad \times \left({\displaystyle{1000{\rm K}} \over {T}}\right)^{3/4} \times \left({\displaystyle{\rho} \over {0.4\,{\rm g}}\,{\rm cm}^{-3}}\right)^{1/2} \times \left({\displaystyle{\Delta R} \over {100\,{\rm cm}}}\right)^{1/2}\,{\rm yrs},} $$

where the thickness of the construction is normalized on 100 cm and ρ is normalized on the density of graphene. As it is clear from Eq. (13), the DS is stable oscillating with the period ~ 33.8 yrs. One can straightforwardly show that for the mentioned temperature the required mass of a megastructure is less than the mass of Earth. On the other hand, as we have already discussed since smaller DSs will require less material, the issue that the civilization might address would be to increase the surface temperature of the sphere.

In Fig. 2 we show the behaviour of P _DS versus the surface temperature for graphene having the melting temperature, 4510 K (Los et al. Reference Los2015). Here we assume that if our civilization (almost Level-I) might produce such materials, a super advanced one is able to do more efficiently. It is clear from the plot, that the period varies from 33.8–0.53 yrs. This result automatically means that the search for Dysonian megastructures could be widened and observations should be performed not only in the infrared spectrum, but in the optical band. One of the characteristic features, that might distinguish the real DS from stars is the luminosity-temperature relation. For example, main sequence stars having temperature of the order of 2000 K belong to low luminosity M stars (Carroll & Ostlie Reference Carroll and Ostlie2010), whereas the megastructure might emit the luminosity equal or even higher than the Solar luminosity. The temperature range we consider is (2000–4000 K), which means that the spectral radiance peaks at wavelengths (725–1450 nm), havinga significant fraction in the optical band. Here we examined graphene as a particular example just to show that if extraterrestrials consider strong materials, their DS might exhibit an interesting behaviour different from normal stars.

Fig. 2. Here we plot the dependence of oscillation period of the DS for 3D Graphene with the density ρ = 0.4 g cm⁻³ and the melting point, T _m ≈ 4510 K.

Another important observational fingerprint follows from the results obtained above. Due to the oscillation of the hot DS its detected flux will be variable, with the period P _DS. From the figure, it is clear that the values of P _DS are typical timescales of the variability of very long period pulsating stars, which normally belong to spectral class: F, M, S or C. But F stars usually have very high temperatures, of the order of 7000 K, typical temperatures of M stars although are relatively low (2400–3700) but their luminosities are almost two orders of magnitude less than the Solar luminosity. Unlike them, S and C stars are highly luminous (in comparison with the Solar luminosity) (Carroll & Ostlie Reference Carroll and Ostlie2010). Therefore, these discrepancies might be good indicators of potentially interesting objects. It is worth noting that the aforementioned examples only show a certain tendency how an observational pattern of the megastructures could be significantly different from the behaviour of variable stars. This in turn, means that an analysis of rich observational data provided by several optical telescopes might be very promising.

The surface of the DS is not an absolutely rigid body and therefore, it might vibrate under the influence of perturbations transversal to the surface, which might be induced either by the radiation pressure or by means of the star's wind. Therefore, another possible source of variability of DSs could be the mechanical waves generated on the 2D surface of the megaconstruction. It is straightforward to show that if a membrane is stressed with a tension per unit length, (along the surface of the DS) τ and its mass per unit area is Σ, the transverse wave speed along the surface is of the order of

(14)$$ \upsilon_{{\rm w}} \approx\left({\displaystyle{\tau \over \Sigma}}\right)^{1/2}. $$

Even if the construction does not envelope the star completely (Dyson Swarm) it still can have similar interesting observational features. In particular, by means of these waves the surface will vibrate exhibiting the variability of emission intensity. Unlike the completely closed surface, the Dyson swarm will experience the gravitational force from the Star leading to the following value of τ

(15)$$ \tau\approx \alpha G {\displaystyle{MM_{{\rm s}}} \over {2\pi R^3}}, $$

where G ≈ 6.67 × 10⁻⁸ Nm² kg⁻² is the gravitational constant, M _s is the star's mass and α is the coefficient, that depends on how complete the DS is (for the closed surface α = 0). We assumed that the tension is caused by gravitation (although the possible rotation of the DS might also influence the value of τ). It is clear that the timescale, P _w, for waves to travel from one point to a diametrically opposite location is given by υ_w/(πR) which for Graphene will be 1.5α yrs for 2000 K and 0.8α yrs for 4000 K. In case of higher harmonics, the corresponding values will be even less. Although we do not know the value of α and the origin of the tension is unclear, the estimate shows that an aforementioned anomalous variability might be a good sign for a potential DS.