The Extended Kardashev Scale

Civilizations can be classified according to the power they produce using the equation

$$\begin{eqnarray}&&{K}_{{\rm{P}}}={\mathrm{log}}_{10}(P/{10}^{6})/10,\end{eqnarray} \tag{ 1 }$$

where K_P is the type of civilization and P is power in watts ($\ne$ 0). The equation yields whole numbers 0, 1, 2... for P = 10⁶, 10¹⁶, 10²⁶... and decimal fractions for intermediate values, resulting in a continuous scale unlike Kardashev's Roman numeral categories. The power P corresponding to a value K_P is simply

$$\begin{eqnarray}&&P={{10}^{10K}}_{{\rm{P}}}{10}^{6}.\end{eqnarray} \tag{ 2 }$$

Equations similar to Equation (1) have been attributed to Sagan, although no equation appears in his book Cosmic Connection (Sagan 1973a), which is often cited as the source. The earliest equation found in a literature search was a Wikipedia article on the Kardashev Scale dated 27 November 2004, more than 30 years after Sagan's book, which was cited as the source. Some equations that have appeared are summarized below, most equivalent to Equation (1) and all citing Sagan (1973a).

Wikipedia gives the equation

$$\begin{eqnarray}&&K=\left({{\rm{log}}}_{10}({\rm{P}})-6\right)/10,\end{eqnarray} \tag{ 3a }$$

where "P is the power it uses, in watts."¹

Piotelat and Cerceau wrote

$$\begin{eqnarray}&&K={{\rm{log}}}_{10}(W)/10,\end{eqnarray} \tag{ 3b }$$

where W is "energy consumption in megawatts" (Piotelat & Cerceau 2013, p. 209).

Wright et al. (2014, p. 2) wrote

$$\begin{eqnarray}&&K={{\rm{log}}}_{10}(P/10\,{\rm{MW}})/10,\end{eqnarray} \tag{ 3c }$$

where P is "the civilization's total energy supply P, measured in units of 10 MW"; the equation yields the expected values using units of 1 megawatt rather than 10.

Ćirković (2015) wrote

$$\begin{eqnarray}&&n=1+1/10\,{{\rm{log}}}_{10}(E/{10}^{16}\,{\rm{W}}),\end{eqnarray} \tag{ 3d }$$

where n is the Kardashev type and E is power in watts.

Table 3 presents some examples for historical terrestrial power consumption and future projections illustrating the usefulness of the scale and its decimal subdivision. Power consumption in the years 1800, 1900, and 2000 yield K_P = 0.58, 0.61, and 0.71, respectively, showing long-term changes in the first decimal place. Power consumption in the recent year 2015 yields K_P = 0.72, showing short-term change from 2000 to 2015 in the second decimal place.

Equation (1) yields K_P = 0.0 for 10⁶ W, a type and power level not considered by Kardashev, Sagan, or Lemarchand, but a natural extension of the 10¹⁰ intervals of the planetary, stellar, and Galactic types suggested by Sagan and Lemarchand. One megawatt is very small compared with the other types that have astronomical scales; it is comparable to, among other things, the metabolic power of the largest terrestrial animals and groups of animals. The blue whale Balaenoptera musculus has a basal metabolic rate of up to 1.9 × 10⁴ W and active metabolic rate up to 4.6 × 10⁵ W (Lockyer 1981)—nearly half a megawatt. The human basal metabolic rate is approximately 10² W and up to 10³ W in bursts (Smil 2017), so thousands of humans might command a megawatt with muscle power alone, but interstellar signaling would not be expected from metabolic power alone. Fossil fuels are an energy source of biological origin accumulated over geological time, and in the terrestrial case accounted for approximately 85% of the 1.7 × 10¹³ W world primary energy consumption in 2015 (BP 2018), which seems like a reason to characterize the lower part of the scale as biological. Earth circa 2015 would be Type 0.72.

The value 10¹⁶ W is taken as a planetary-scale power level, following Sagan's suggestion for a Type 1.0 at 10¹⁶ W; Kardashev had no comparable power level and his category I was based on terrestrial power production in 1964, which is no longer useful. The value 10¹⁶ W is roughly comparable to terrestrial insolation of 1.7 × 10¹⁷ W based on the solar constant 1361 W m⁻² (Prša et al. 2016) and the area of the planet illuminated. Insolation seems like a natural way to think about power at a planetary scale because it can be defined using astronomical information, although planets vary in insolation, and energy is available from many other sources including fossil fuels, hydro, and nuclear. Insolation might be a very rough upper limit to the total power that can be consumed on a planet without causing thermal environmental problems (Rebane 1993). Terrestrial power production in the recent year 2015 was 1.7 × 10¹³ W from Table 3 which is only 10⁻⁴ of terrestrial insolation and 1/588 of the categorical 10¹⁶ W.

A Type 1.0 civilization would be capable of an isotropic broadcast over 10³ ly requiring ∼10¹⁵ W, assuming a search system using a 100 m antenna comparable to the one currently used at the Green Bank Telescope (GBT)—if it dedicated 10% of its energy resources to that purpose, which suggests that isotropic broadcasts require civilizations near that type and likely higher. Highly directional transmissions over 10³ ly, on the other hand, are possible for a civilization at our current level—for example requiring 10⁷ W with 300 m antennas on both ends, or 10 times the power of the Arecibo planetary radar (Campbell et al. 2002).

Table 3 includes terrestrial power projections for future years, P_t, calculated for time, t, assuming a constant growth rate, r, using

$$\begin{eqnarray}&&{P}_{t}={P}_{0}{\left(1+r\right)}^{t}\end{eqnarray} \tag{ 4 }$$

where P₀ is the value at the starting time. Using Kardashev's 1% growth rate and starting from 2015, terrestrial power production would reach the planetary level 1.0 in the year 2654. Actual energy growth rates are not constant, so such projections are not predictions; the actual growth rate for the years 2006–2016 was 1.7% (BP 2018), and rates such as 2.6% have been used in some earlier projections (e.g., Zubrin 1999).

The value 10²⁶ W is taken as a stellar-scale power level, following Sagan's suggestion. That value differs from Kardashev's 4 × 10²⁶ W value and from the 3.828 × 10²⁶ W luminosity of our Sun (Prša et al. 2016) by a factor of four, but it is broadly representative of the F, G, and K class stars often considered as potentially habitable. The luminosity of M stars is several orders of magnitude smaller, but Sun-like is the class known to have life and therefore possibly technosignatures. A civilization commanding power equivalent to the luminosity of our Sun would be Type 2.06 rather than Type II on the original scale.

Schemes have been suggested for capturing stellar luminosity other than on the surface of a planet, such as solar power satellites, space settlements (O'Neill 1979), and Dyson spheres or swarms (Dyson 1960). Capturing the full luminosity of a star is difficult to envision, but even higher rates of energy production have been imagined, for example by extracting and fusing hydrogen from a star or gas-giant planets (Criswell 1985). The approximate output of a Sun-like star seems most useful for present purposes, and the decimal scale supports distinctions such as K_P = 1.8 for capturing 1% of the categorical stellar 10²⁶ W or producing the power in other ways.

Human energy production would reach the Type 2.0 stellar scale in the year 4968 at a 1% annual growth rate from 2015, from Table 3. Kardashev projected 3200 years from 1964 at that rate, which would be the year 5164; the 196-year difference in projections is due to adopting Sagan's value for the stellar level, using 2015 as the base year, and Kardashev rounding to the nearest century.

Isotropic broadcasts detectable across the Galaxy become possible between the planetary and stellar scales, requiring for example ∼10¹⁸ W for 50,000 ly assuming a 100 m search system comparable to one used at the GBT. A few searches for evidence of infrared signatures from stars possibly due to Dyson-like structures have been carried out, including one covering 96% of the sky with sensitivity sufficient to detect a Dyson sphere with the luminosity of the Sun out to 300 pc (Carrigan 2009). Searches for evidence of megastructures are also possible in transit data from exoplanet searches (Wright et al. 2016).

The value 10³⁶ W is taken as a Galactic-scale power level, following Sagan's suggestion. One estimate of the total luminosity of the Milky Way Galaxy is 2.3 × 10¹⁰ times solar luminosity (Freudenreich 1998) or 8.8 × 10³⁶ W, and another estimate is 6.7 × 10¹⁰ times solar luminosity (Kent et al. 1991) or 2.6 × 10³⁷ W. A civilization commanding power equivalent to the luminosity of the Milky Way would be Type 3.1 using either estimate (rounding to one decimal place) rather than Type III on the original scale. Taking the Galactic scale as approximately one order of magnitude smaller than the Milky Way estimate fits the 10¹⁰ intervals, and seems reasonable because the Milky Way is larger than most galaxies—second largest in the Local Group of dozens.

This category does not assume any particular scenario such as a single civilization colonizing a galaxy rather than multiple independently evolved civilizations, or Dyson structures surrounding all or many stars, or power being generated from a supermassive black hole (Inoue & Yokoo 2011), but the waste heat of energy use at such a large scale might be detectable regardless of how it is produced. One search of ∼10⁵ other galaxies for galaxy-spanning civilizations found no evidence of "an alien civilization reprocessing more than 85% of its starlight into the MIR" (mid-infrared) in data from the Wide-field Infrared Survey Explorer (WISE) telescope survey (Griffith et al. 2015, p. 25), with one problem being the need to identify and exclude intrinsically dusty galaxies with infrared signatures. Another analysis found no obvious evidence of a civilization using the bulk of a galaxy's starlight for its own purposes in a sample of 137 galaxies (Annis 1999).

Kardashev did not define a Type IV for power comparable to the luminosity of the observable universe, but adding such a level at 10⁴⁶ W is consistent with the other 10¹⁰ intervals and is within two orders of magnitude of current estimates for the luminosity of the observable universe. One such estimate is 10⁴⁸ W, assuming 10¹¹ galaxies each averaging 10¹¹ stars with an average luminosity of a few times 10³³ erg s⁻¹ like the Sun (Wijers 2005). The Hubble Ultra Deep Field observations reported more than 10⁴ galaxies in a 11' field (Beckwith et al. 2006), which is consistent with a total of 10¹¹ galaxies, and a more detailed analysis reported the "number of galaxies currently detectable within the universe with a hypothetical all-sky Hubble Space Telescope survey at UDF-Max depth" as 2.47 × 10¹¹ (Conselice et al. 2016, p. 16). Not all galaxies can be detected with current instruments; an estimated 2 × 10¹² galaxies "in principle could be observed" (Conselice et al. 2016, p. 12). The type for a civilization spanning the observable universe might very well exceed 4.0.

Sagan wrote "there is no provision for a Type IV civilization, which by definition talks only to itself" (Sagan 1973a, p. 234), but the decimal scale accommodates more complex scenarios such as one or many civilizations spanning various fractions of the observable universe whose radio, optical, or infrared emission we might detect if any exist.

Sagan proposed using the 26 letters of the English alphabet to classify the amount of information stored by a civilization, with A = 10⁶ bits, B = 10⁷ bits, and so on, and he wrote that terrestrial civilization circa 1973 could be "well characterized by something like 10¹⁴ or 10¹⁵ bits" based largely on estimated holdings of the largest libraries (Sagan 1973a, p. 237). He described a composite terrestrial power and information type as 0.7 H (H actually corresponds to 10¹³ bits).

An equation similar to the one used for power can be used for an information scale K_I,

$$\begin{eqnarray}&&{K}_{I}={\mathrm{log}}_{10}(I/{10}^{6})/10,\end{eqnarray} \tag{ 5 }$$

where I  = information in bits ($\ne$ 0). Sagan's estimate of 10¹⁴ or 10¹⁵ bits circa 1973 yields K_I = 0.8 or 0.9.

The constant 10⁶ is the same as in the power equation and yields K_I = 0.0 for 10⁶ bits, which is also the value Sagan chose for his category "A," saying that 10⁶ bits were less than "any human society that we know well—and a good beginning point" (Sagan 1973a, p. 237). Robertson (1998) estimated 10⁷ bits as available to individual humans before language and writing were developed, and Landauer estimated the "functional information content of human memory" as 10⁹ bits at midlife based on testing of text and image retention (Landauer 1986, p. 491).

Estimates of the information store of human civilization vary depending on what is counted and when. Kardashev (1964) estimated 10¹⁴ bits circa 1964 counting 10⁸ publications at 10⁶ bits each. Lesk (1997) estimated 10¹⁸ bits in 1997 including books, maps, films, and sound recordings. Robertson (1998) estimated 10¹⁷ bits available in books and other publications in 1998 and 10²⁵ bits including computers. Lyman & Varian (2000) estimated 10¹⁸–10¹⁹ bits being added annually around 2000, including paper, film, optical, and magnetic storage with a growth rate of 50%. Taking the terrestrial value as 10¹⁸ bits in 1998 and using a modest 1% growth rate results in approximately 1.2 × 10¹⁸ bits in the year 2015 and K_I = 1.2; using the 50% growth rate results in approximately 10²¹ bits in the year 2015 and K_I = 1.5.

A similar equation can be used to classify the population of a civilization K_N,

$$\begin{eqnarray}&&{K}_{N}={\mathrm{log}}_{10}(N)/10,\end{eqnarray} \tag{ 6 }$$

where N = population ($\ne$ 0), omitting the 10⁶ scaling because population is a count of individuals. A population of 1 yields K_N = 0.0, and the world population of 7.38 × 10⁹ in 2015 (U.N., 2017) yields K_N = 0.99.

World population forecasts for the year 2100 are 7.3, 11.2, and 16.5 billion for low, medium, and high variants (U.N., 2017) yielding K_N = 0.99, 1.00, and 1.02, respectively. A value of K_N = 2.0 would correspond to a population of 10²⁰, which could describe a galactic-scale civilization with 10¹⁰ planets each having a population of 10¹⁰, or many other scenarios.

A similar equation can be used to classify the mass of a civilization's constructions K_C,

$$\begin{eqnarray}&&{K}_{C}={\mathrm{log}}_{10}(C/{10}^{6})/10,\end{eqnarray} \tag{ 7 }$$

where C is mass of constructions in metric tonnes ($\ne$ 0). One estimate for total human construction is 3 × 10¹³ t (Zalasiewicz et al. 2017), which yields K_C = 0.75, a mass comparable to a solid sphere approximately 30 km in diameter with the density of aluminum or concrete. Type 1.0 would correspond to 10¹⁶ t, comparable a solid sphere several hundred kilometers in diameter like the asteroid Juno. Kardashev (1985) speculated about the possibility of extremely large "superstructures" that might be detectable but did not suggest a classification scheme.

The Earth's composite type classified by power, information, population, and mass of constructions in 2015 could be expressed as K_P,I,N,C = 0.72, 1.50, 0.99, 0.75.

The power available to a civilization has obvious implications for the detectability of extraterrestrial intelligence, because power is required for signaling, and because very large-scale power consumption might be detectable as thermal emission without intentional signaling. The large mass of constructions might be relevant, because very large structures might be detected by astronomical observations, as in speculation that variations in brightness of KIC 8462852 might be due to large-scale engineering (Boyajian et al. 2018), Dyson spheres or swarms, and Kardashev's superstructures. Population might be relevant as driving growth, although not directly observable over interstellar distances. The information store of civilizations would not be directly observable over interstellar distances, but conceivably could be detected if transmitted, because as Kardashev (1964) noted the ∼10¹⁴ bits of a terrestrial 1964 civilization could be transmitted across the Galaxy using a 10⁹ bit s⁻¹ bandwidth channel for approximately one day.