Mathematics of the Kardashev Scale

Short science fiction by Torn MacAlester

The Kardashev scale is a measure of the technological achievement of a civilization.  Traditionally, there have been three levels on the Kardashev scale: Type 1 - corresponding to a planetary civilization, Type 2 - corresponding to a stellar civilization, and Type 3 - corresponding to a galactic civilization. The types of civilization correspond to the amount of power the civilization wields.  Here is the mathematics of the Kardashev scale.

The Kardashev scale is defined by the expression:

K(P)=(log P - 6)/10

where P is the power generated by the civilization expressed in Watts. For planetary civilizations P=1016 W, K(P)=1 (e.g. Type I).  For stellar civilizations  P=1026W, K(P)=2 (e.g. Type II).  For galactic civilizations P=1036 W, K(P)=3 (e.g. Type III).  Earth is currently at K(P)~0.73, lower than the threshold for Type I.

Another useful scale is the power available to the individual.  For example, we can set:

 KI(P)=K(P) - (log I) / 10

where I is the number of individuals in the civilization. This provides another measure of the mathematics of the Kardashev scale.  If we assume that a Type I civilization has a population of 10 billion individuals, KI(P)=0. In essence, we see that there is a MW of power per individual for a Type I civilization that has approximately Earth’s population.  By comparison, the earth has a value of about -0.298, more population than power.

A good discussion of the Kardashev scale can be found at Atomic Rockets web site.

Another resources is the Wikipedia page on the Kardashev scale.