Main considerations

In this section we consider the possibility of existence of extraterrestrial micro replicators, study their physical characteristics and observational fingerprints.

We assume that a probe with a lengthscale, r, moves with the constant velocity υ = βc (β < 1) in a medium with the number density, n. In order to replicate, the probe should collect a material during the motion. By taking into account that an effective cross-section of the robot is A = αr ² the mass rate writes as

(1)$$\displaystyle{{{\rm d}M} \over {{\rm d}t}} = \alpha \beta r^2m_0nc,$$

where m ₀ is the mass of a molecule, c is the speed of light and for a spherical shape of the probe (henceforth we use the spherical shape of robots) with radius r one has: α = π. By assuming that the mass of the probe is M = γξr ³ρ, the timescale of replication should be of the order of

(2)$$\eqalign{\tau & = \displaystyle{M \over {{\rm d}M/{\rm d}t}} = \displaystyle{{\gamma \xi } \over {\alpha \beta }} \times \displaystyle{\rho \over {m_0n}} \times \displaystyle{r \over c}\simeq 6.2 \times \displaystyle{\xi \over {0.1}} \cr & \times \displaystyle{{0.01} \over \beta } \times \displaystyle{\rho \over {0.4{\rm g}{\mkern 1mu} {\rm c}{\rm m}^{-3}}} \times \displaystyle{{{10}^4{\mkern 1mu} {\rm c}{\rm m}^{-3}} \over n} \times \displaystyle{r \over {0.1{\mkern 1mu} {\rm \mu} \,{\rm m}}}{\mkern 1mu} {\rm years},} $$

where for the spherical shape γ = 4π/3 and ξ is the fraction of the total volume filled with the material the probe is made of, ρ is normalized by the density of the Graphene (as an example we have considered – up to now the strongest material), n is normalized by typical values of molecular clouds.

If the number of initial probes is N ₀, then by assuming that the time of replication equals τ, then the time evolution of the total number of replicators writes as

(3)$$ N(t) = N_0\times 2^{t/\tau}. $$

In Fig. 1 we show the behaviour of N versus time for three different values of β. The set of parameters is: N ₀ = 100, ξ = 0.1, m ₀ = m _H (m _H is the mass of the Hydrogen atom), ρ = 0.4 g cm⁻³, n = 10⁴ cm⁻³. The results have been presented for the parameters of the HII interstellar region of atomic Hydrogen (Carroll and Ostlie Reference Carroll and Ostlie2010). As it is clear from the plots the total number of probes might increase extremely rapidly ‘invading’ the whole region of an interstellar cloud. In the framework of the paper we do not go into details how the probes are accelerating up to the very high non-relativistic velocities: β = {0.01;0.02;0.1}, but one can show that from the point of view of energetics maintenance of velocity might be quite realistic. In particular, during the motion in the interstellar medium the probes will encounter particles (Hydrogen atoms) leading to the power of an engine providing the propulsion system

(4)$$P\simeq \upsilon ^2\displaystyle{{{\rm d}M} \over {{\rm d}t}}.$$

On the other hand, the captured mass rate will lead to the following total energy rate

(5)$$\displaystyle{{{\rm d}E} \over {{\rm d}t}} = c^2\displaystyle{{{\rm d}M} \over {{\rm d}t}}.$$

Since we consider the non-relativistic probes with υ/c ≪ 1, the propulsion system will require a tiny fraction of the total gained mass to convert into energy, which we assume could not be a problem for a Type-II civilization.

Fig. 1. The behaviour of N versus time for three different values of β = {0.01;0.02;0.1}. The set of parameters is: N ₀ = 100, ξ = 0.1, m ₀ = m _H (m _H is the mass of the Hydrogen atom), ρ = 0.4 g cm⁻³, n = 10⁴ cm⁻³.

If such an ‘invasion’ will take place the Von-Neumann replicators should have interesting observational consequences. In particular, as it is clear from Fig. 1, the number of probes after crossing the distance of the order of 1pc increases from 100 up to ~6 × 10³³. This means that if the replicators always stay in one plane surface – ‘the front wave of probes’ – then the corresponding lengthscale of the surface $\ell \sim \sqrt {\pi r^2 N(t)}$ after 650 years for β = 0.01, after 325 years for β = 0.02 and after 65 years for β = 0.1 becomes of the order of 10¹² cm, when the initial lengthscale was 10¹⁶ orders of magnitude less. The total mass of the probes will be of the order of M ~ 4πξρ/3r ³ N(t) ~ 10¹⁸ g (ξ ≃ 0.1) which is a typical mass of a comet having a lengthscale of several kilometres. The probes encountering and collecting the protons will accelerate them with the acceleration, a ~ υ²/(2κr), where κ ≤ 1 represents the dimensionless lengthscale of acceleration. This in turn will inevitably lead to the emission power of a single probe

(6)$$L_0\simeq \displaystyle{2 \over 3}\displaystyle{{e^2a^2} \over {c^3}},$$

where e is the electron's charge. By taking into account that the average time of acceleration is of the order of 2rκ/υ the total emitted energy of a probe corresponding to a single event writes as

(7)$${\rm \epsilon }_0\simeq L_0 \times \displaystyle{{2r\kappa } \over \upsilon }\simeq \displaystyle{{e^2\beta ^3} \over {3r\kappa }}.$$

Therefore, the average radiation luminosity of a single probe resulting from the multiple encounters with the interstellar protons is given by

(8)$$L_{\rm p}\simeq \displaystyle{\pi \over {3\kappa }}nrce^2\beta ^4,$$

which leads to the total luminosity of the system of self-reproducing Von-Neumann automata

(9)$$L_{{\rm tot}}\simeq \displaystyle{\pi \over {3\kappa }}nrce^2\beta ^4N_0 \times 2^{t/\tau }.$$

In Fig. 2 we show the behaviour of L _tot versus time. κ = 1 and the other set of parameters is the same as in Fig. 1. As it is evident from the plots the luminosity very rapidly increases up to considerably high values of the order of 10¹⁵⁻⁻²⁰ erg/s which might be visible by telescopes. On the logarithmic scale the plots are represented by straight lines, which means that the increment, Γ, characterizing the exponential increase of the luminosity

(10)$$ L_{\rm tot}\simeq L_{\rm p}e^{\Gamma t} $$

is a constant value. Indeed, from equations (2) and (10) one derives

(11)$$\eqalign{\Gamma & = \displaystyle{{\alpha \beta } \over {\gamma \xi }} \times \displaystyle{{m_0n} \over \rho } \times \displaystyle{c \over r} \times \ln 2\simeq 0.11 \times \displaystyle{{0.1} \over \xi } \times \displaystyle{\beta \over {0.01}} \cr & \times \displaystyle{{0.4{\mkern 1mu} {\rm g}{\mkern 1mu} {\rm c}{\rm m}^{-3}} \over \rho } \times \displaystyle{n \over {{10}^4{\mkern 1mu} {\rm c}{\rm m}^{-3}}} \times \displaystyle{{0.1{\mkern 1mu} \mu {\rm m}} \over r}{\mkern 1mu} {\rm year}{\rm s}^{-1}.} $$

As one can see from this expression the luminosity increment is a linearly increasing function of the velocity, which is a natural result because the number of incident protons is a linear function of υ (see equation (1)). The corresponding timescale is of the order of 9 years and for the highest considered velocity, 0.1 c the process of luminosity increment is ten times efficient and the corresponding timescale is 0.9 years.

Fig. 2. Dependence of L _tot on time for three different values of β: 0.01,0.02,0.1. κ = 0.1 and the other set of parameters is the same as in Fig. 1.

From the expression of Γ it is clear that the corresponding value is inverse proportional to the size of the probe and for higher values of r the increment will be smaller. On the other hand, self-reproducing micro robots are more efficient compared to macro probes. In particular, if one considers a robot with a size of the order of 1 cm on the same distance 1 pc the increase in mass of a single probe will be less than 0.1%, whereas the micro robots on the same distance ‘invade’ a huge area of a cloud. One can also define the maximum size of the probe, which still can be used for the replication. It is clear that the process is efficient if N exceeds the initial value at least ten times, then from equation (3) one obtains the estimate of the maximum size of the probe

(12)$$\eqalign{& r_{\max }\simeq \displaystyle{{\ln 2} \over {\ln 10}} \times \displaystyle{\alpha \over {\gamma \xi }} \times \displaystyle{{m_0n} \over \rho } \times D\simeq 3.17 \times 10^{-4} \cr & \times \displaystyle{{0.1} \over \xi } \times \displaystyle{{0.4{\mkern 1mu} {\rm g}{\mkern 1mu} {\rm c}{\rm m}^{-3}} \over \rho } \times \displaystyle{n \over {{10}^4{\mkern 1mu} {\rm c}{\rm m}^{-3}}} \times \displaystyle{D \over {2pc}}{\rm cm},} $$

where D is the size of the cloud. The aforementioned formula gives a definition of a macro size, after which the process of replication is very inefficient.

Therefore, type-II extraterrestrials would prefer to use micro robots than large-scale macro probes. Still there is a possibility to make the process of reproduction efficient, but for that the probes need to land on rocky planets. This compared to the continuous process of replication of micro devices in interstellar clouds seems to be less efficient because landing on a planet and fleeing from it require special manoeuvring.

Another interesting issue which can be addressed concerns the minimum size of the replicators. In particular one can see from equation (10) that the total luminosity exponentially increases and we have already discussed that small size probes give higher increment. On the other hand, Type-II civilization is by definition a society which can utilize the power of the order of the Solar luminosity $L_{\odot }\simeq 3.9\times 10^{33}$ erg s ⁻¹. By imposing on equation (10) the constraint $L_{\rm tot}\leq L_{\odot }$ one can straightforwardly derive the inequality

(13)$$\ln 2 \times \displaystyle{\alpha \over {\gamma \xi }} \times \displaystyle{{m_0n} \over \rho } \times \displaystyle{D \over r} \le \ln \left( {\displaystyle{{L_\odot } \over {L_p(r)}}} \right),$$

where L _p(r) is given by equation (8). By assuming the parameters applied to Fig. 1 one can numerically show that the minimum value of r corresponding to Type-II civilization is of the order of 7 × 10⁻⁶ cm. If the size is less than this, the alien society must be even more advanced consuming much higher energies.

One of the factors which influence the luminosity increment is the density of the interstellar cloud which usually change from location to location. It is observationally confirmed that the density might change from a few particles per cm³ up to 10⁶ per cm³ (Carroll and Ostlie Reference Carroll and Ostlie2010), which might drastically increase the luminosity growth rate.

As to the spectral picture of radiation, the characteristic frequency is of the order of inverse of value of the acceleration timeFootnote ¹

(14)$$\nu \simeq \displaystyle{\upsilon \over {2r\kappa }}\simeq \displaystyle{{1.5 \times {10}^{13}} \over \kappa } \times \displaystyle{\beta \over {0.01}} \times \displaystyle{{1{\mkern 1mu} \mu \,{\rm m}} \over r}{\rm Hz}.$$

In the previous calculations we have used κ = 1, in which case the self-replicating probes will be visible in the infrared spectral band. But in fact the acceleration of protons might take place in much shorter distances, leading to even X-rays.

Apart from the HII regions there are also the molecular clouds in the Milky Way, where the number density varies in the interval (10² −10⁶) cm⁻³ (Carroll and Ostlie Reference Carroll and Ostlie2010), implying that the process of replication might be as efficient as in HII. During the process of encounter of Hydrogen molecules, H ₂, with respect to a probe they will have kinetic energy

(15)$$K\simeq \displaystyle{{2m_pv^2} \over 2}\simeq 5 \times 10^8{\mkern 1mu} {\rm eV},$$

exceeding by several orders of magnitude the dissociation energy (~1 eV) of the molecule as well as the ionisation energy of the Hydrogen atom, 13 eV. This means that likewise the previous case the particles will emit quite efficiently, which potentially might be detectable.