Mathematics of the Kardashev Scale

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℗=(log P — 6)/10

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

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

 K_I℗=K℗ — (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, K_I℗=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.