Puny Earth doomed?

Calculus Level 4

A race of super intelligent aliens have very large perfectly spherical heads. Each kilogram of their brain operates at the maximum possible computational speed (Bremermann's limit) which is

\[\frac { { c }^{ 2 } }{ h } \approx 1.359\times10^{50}\text{ bits}\cdot\text{s}^{-1}\cdot\text{kg}^{-1}.\]

The radius of an alien's head is \(0.5\text{ m}.\) Since the density of its brain is non-uniform, it is given by

\[\rho (r)=\frac { 0.5-r }{ 10^{ 3 } }\text{ kg/m}^3,\]

where \(r\) is the distance (in meters) from the center.

A single human brain is capable of computing \(400\times10^9\) bits per second. In terms of computing power, how many human brains is a single alien brain equivalent to?

Details and Assumptions

  • For this problem the given approximation of Bremermann's limit will suffice.

  • Take a look at Micheal Steven's cool YouTube video "How many things are there?" where he uses Bremermann's limit to count everything.


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