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.