Solar Emission in the ISM band

The sun is 6.955e8 meters in diameter, and 1.496e11 meters away. Its effective black body temperature is 5778k.

The Planck black body spectrum is:

\Large { { 2 h {\nu}^3 } \over { c^2 ( e^{h \nu / k T } - 1 ) } } Watts / steradian m² Hz

For h \nu << kT , this approximates to:

\large { { 2 h {\nu}^3 } \over { c^2 ( 1 + h \nu / k T - 1 ) } }

\large { { 2 h {\nu}^3 } \over { c^2 h \nu / k T }}

\large { { 2 {\nu}^2 k T } \over { c^2 } }

{ 2 kT / {\lambda}^2 }

Multiplied by half the sky, 2 \pi steradians, and the bandwidth BW , the power per square meter is

4 \pi kT ~ BW / {\lambda}^2 W / m²

Multiplied by the surface area of the sun 4 \pi {Rsun}^2 :

( 4 \pi Rsun / \lambda )^2 ~ kT ~ BW Watts

For the the ISM band, \lambda = 2.997e8 m / 5.8 GHz = 0.0517 meters and BW = 15 MHz. The Boltzmann constant k is 1.3806488 × 10-23 J/K so kT is 7.977E-20 Joules . 4 \pi Rsun / \lambda is 4 π 6.955e8 / 0.0517 = 1.69e11 (unitless). The power emission from the entire Sun in ISM is 2280 Watts / Hz or 34.2 GW for the entire ISM band.

Brighter than 1000 suns

If we tried to make 100 Terawatts of earth energy from space solar power (10kW / person, 10 billion people), and broadcast 120 Terawatts of ISM band power to do it, the brightness of the constellation would be 3500 times the entire sun radiating into all of space. If the energy was concentrated into a 1 Hz band, it could outshine the mostly-red-dwarf galaxy in that narrow band.

How much hits the earth? The earth's disk is π Re², Re =6370 km, and the sun is Ro = 1.496e8 km away, irradiating a sphere with an area of 4 π Ro². The fraction reaching earth is 0.25 (Re/Ro)² or 4.54E-10 of the sun's output, 15.5 Watts for the entire ISM band, or 1 μW / Hz . It is sobering to realize that a single wireless access point can generate more power in its channel than the Sun delivers to the whole planet.