A Share of the Sky

solved Units

Q. How much of the sky does a patch of area $A$ cover, seen from $r$ away?

Inputs

area in square metres

A

distance

r

2.00 m

Outputs

solid angle in steradians

Ω

How it's solved

Solved with Units

Given

$A$ , $r$ yours to vary

Find $Ω$ .

Working

area over radius squared $Ω = \frac{A}{r^{2}}$
$Ω = \frac{4}{2^{2}}$
$Ω = 1$

Conclusion

Its area over the square of how far away it is: Ω = A ⁄ r². Like the radian, the steradian is a pure number, and the whole sky around a point is 4π of them.

The web of concepts

given read from the words equation asked

The Ruby that solves it

The question and its solution are small Ruby files. The engine runs them.

question.rb

1require_relative "../physics"23question :share_of_the_sky do4  describes "%{solid_angle} does a patch of area %{area} cover, seen from %{radius} away?"5  given :area6  given :radius7  asks solid_angle: "How much of the sky"8end

solution.rb

1require_relative "../questions/share_of_the_sky"2require_relative "../science/units_and_measurement/units"34solve :share_of_the_sky, using: Units do5  take area: :A, radius: :r6  take solid_angle: :Omega7  find_value :solid_angle89  conclusion "Its area over the square of how far away it is: Ω = A ⁄ r². Like the radian, the " \10             "steradian is a pure number, and the whole sky around a point is 4π of them."11end

playground.rb

1require_relative "../solutions/share_of_the_sky"23play_with :share_of_the_sky do4  title "A Share of the Sky"56  input :A, 0.0..100.0, step: 0.5, default: 4.0, as: "area in square metres"7  input :r, 0.5..10.0, in: "m", step: 0.1, default: 2.0, as: "distance"89  output :Omega, as: "solid angle in steradians"10end

The laws it drew on

quantity	written	units	within	from
speed of light	$c$	m/s	—	Units
avogadro constant	$N_{A}$	mol⁻¹	—	Units
length	$l$	m	0 … ∞	Units
mass	$m$	kg	—	Units
time	$t$	s	0 … ∞	Units
electric current	$I$	A	—	Units
temperature	$T$	K	—	Units
amount of substance	$n$	mol	—	Units
luminous intensity	$I_{v}$	cd	—	Units
arc length	$s$	m	—	Units
radius	$r$	m	0 … ∞	Units
plane angle	$θ$	—	—	Units
area	$A$	m²	—	Units
solid angle	$Ω$	—	—	Units
light time	$t_{light}$	s	—	Units
light distance	$d_{light}$	m	—	Units
number of entities	$N$	—	—	Units
baseline	$B$	m	—	Units
distance away	$D$	m	—	Units
parallax angle	$θ_{p}$	—	—	Units

θ = \frac{s}{r}

arc over radius

Ω = \frac{A}{r^{2}}

area over radius squared

d_{light} = c \cdot t_{light}

distance at light speed

N = n \cdot N_{A}

entities in moles

θ_{p} = \frac{B}{D}

baseline over distance