Shine back at the stars

Today's post is just a fun calculation. Sadly, that means no polemics.

So I was walking home last night, looking up at the stars shining. I wondered how much I'm shining back?

I started by assuming I'm a black body radiating at human body temperature.

T = 37 °C

I considered just my head i.e. a sphere with circumference of an average male human head.

C = 57 cm

I found the surface area of the idealized sphere representing my head.

A = 4πr² = 0.103418882 m²

I plugged these into the Stefan-Boltzmann law to find the photon flux from my head.

N = (1.5205 * 10¹⁵ m^-2 s^-1 K^-3) * T³ * A = 4.691 * 10²¹ photons per second

I looked up a number I'm comfortable with for the distance to the furthest naked-eye visible star.

D = 16308 light-years

And I assumed these stars have on average a similar diameter to our Sun.

d = 1.3914 billion meters

I found the angular diameter of this idealized far-away star.

𝛿 = (206265) * (d / D) = 3.03362676 * 10^-7 arcseconds = 5.167 * 10^-10 degrees

And converted this angle into square degrees.

Ω = πr² = π * (𝛿/2)² = 5.5771095 * 10^-21

Finally I used this solid angle to find the average rate of photons intercepted by the idealized star.

N / (41253 / Ω) = 0.0238439 photons/s = one photon every 42 seconds

So spend a about a minute looking at the stars, and you're shining back at them. At least a little bit.