I forgot to mention another source of difficulty in getting the energy efficiency of the computation down to Landauer’s limit at the CMB temperature.
Recall that the Stefan Boltzmann equation states that the power being emitted from an object by thermal radiation is equal to P=A⋅ϵ⋅σ⋅T4. Here, P stands for power, A is the surface area of the object, ϵ is the emissivity of the object (ϵ is a real number with 0≤ϵ≤1),T is the temperature, and σ is the Stefan-Boltzmann constant. Here, σ≈5.67⋅10−8⋅W⋅K−4⋅m−2.
Suppose therefore that we want a Dyson sphere with radius r that maintains a temperature of 4 K which is slightly above the CMB temperature. To simplify the calculations, I am going to ignore the energy that the Dyson sphere receives from the CMB so that I obtain a lower bound for the size of our Dyson sphere. Let us assume that our Dyson sphere is a perfect emitter of thermal radiation so that ϵ=1.
Earth’s surface has a temperature of about 300K. In order to have a temperature of 4K, our Dyson sphere needs to receive (1/75)4 the energy per unit of area. This means that the Dyson sphere needs to have a radius of about (754)1/2=752=5625 astronomical units (recall that the distance from Earth to the sun is 1 astronomical unit).
Let us do more precise calculations to get a more exact radius of our Dyson sphere.
r2=P4πσ(4K)4, so r=√P4πσ(4K)4=12(4K)2⋅√Pπσ≈1.45⋅1015m which is about 15 percent of a light-year. Since the nearest star is 4 light years away, by the time that we are able to construct a Dyson sphere with a radius that is about 15 percent of a light year, I think that we will be able to harness energy from other stars such as Alpha Centauri.
The fourth power in the Stefan Boltzmann equation makes it hard for cold objects to radiate heat.
I forgot to mention another source of difficulty in getting the energy efficiency of the computation down to Landauer’s limit at the CMB temperature.
Recall that the Stefan Boltzmann equation states that the power being emitted from an object by thermal radiation is equal to P=A⋅ϵ⋅σ⋅T4. Here, P stands for power, A is the surface area of the object, ϵ is the emissivity of the object (ϵ is a real number with 0≤ϵ≤1),T is the temperature, and σ is the Stefan-Boltzmann constant. Here, σ≈5.67⋅10−8⋅W⋅K−4⋅m−2.
Suppose therefore that we want a Dyson sphere with radius r that maintains a temperature of 4 K which is slightly above the CMB temperature. To simplify the calculations, I am going to ignore the energy that the Dyson sphere receives from the CMB so that I obtain a lower bound for the size of our Dyson sphere. Let us assume that our Dyson sphere is a perfect emitter of thermal radiation so that ϵ=1.
Earth’s surface has a temperature of about 300K. In order to have a temperature of 4K, our Dyson sphere needs to receive (1/75)4 the energy per unit of area. This means that the Dyson sphere needs to have a radius of about (754)1/2=752=5625 astronomical units (recall that the distance from Earth to the sun is 1 astronomical unit).
Let us do more precise calculations to get a more exact radius of our Dyson sphere.
r2=P4πσ(4K)4, so r=√P4πσ(4K)4=12(4K)2⋅√Pπσ≈1.45⋅1015m which is about 15 percent of a light-year. Since the nearest star is 4 light years away, by the time that we are able to construct a Dyson sphere with a radius that is about 15 percent of a light year, I think that we will be able to harness energy from other stars such as Alpha Centauri.
The fourth power in the Stefan Boltzmann equation makes it hard for cold objects to radiate heat.