A Share of the Sky

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

What the draft establishes

Fills in once a draft is checked.

The web of concepts

given read from the words equation asked

The equations it used

None yet.

Check an answer

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

The playground

Inputs

area in square metres

A

distance

r

2.00 m

Outputs

solid angle in steradians

Ω

The solution

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 Ruby

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

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

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

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