The Density of the Sun

solved Dimensions

Q. The Sun has a mass of about 2.0 × 10³⁰ kg $m = 2.0 \times 10^{30}$ kg and a radius of about 7.0 × 10⁸ m $r = 7.0 \times 10^{8}$ m. What is its average density, in kilograms per cubic metre?

Inputs

mass

m

2e+30 kg

radius

r

700000000.00 m

Outputs

density

ρ

How it's solved

Solved with Dimensions

Given

$m = 2.0 \times 10^{30}$ kg
$r = 7.0 \times 10^{8}$ m

Find $ρ$ .

Working

volume of a sphere $V = \frac{4 \cdot π \cdot r^{3}}{3}$
$V = \frac{4 \cdot 3.14159265 \cdot {(7 \times 10^{8})}^{3}}{3}$
$V = 1.43675504 \times 10^{27}$ m³
mass per volume $ρ = \frac{m}{V}$
$ρ = \frac{2 \times 10^{30}}{1.43675504 \times 10^{27}}$
$ρ = 1392.02574$ kg/m³
$ρ \approx 1.4 \times 10^{3}$ kg/m³to 2 significant figures

Conclusion

More than water, far less than rock. The Sun is gas, but gas squeezed hard by its own weight.

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 :density_of_the_sun do4  kind :numerical5  describes "The Sun has a mass of about %{mass} and a radius of about %{radius}. %{density}, in kilograms per cubic metre?"6  given mass: "2.0e30 kg"7  given radius: "7.0e8 m"8  asks density: "What is its average density"9end

solution.rb

1require_relative "../questions/density_of_the_sun"2require_relative "../science/units_and_measurement/dimensions"34solve :density_of_the_sun, using: Dimensions do5  take mass: :m, radius: :r6  take density: :rho7  answer find_value(:density, figures: 2)89  conclusion "More than water, far less than rock. The Sun is gas, but gas squeezed hard by its " \10             "own weight."11end

playground.rb

1require_relative "../solutions/density_of_the_sun"23play_with :density_of_the_sun do4  title "The Density of the Sun"5end

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
pi	$π$	—	0 … ∞	Dimensions
gravitational constant	$G$	N m² kg⁻²	—	Dimensions
volume	$V$	m³	—	Dimensions
speed	$v$	m/s	—	Dimensions
acceleration	$a$	m/s²	—	Dimensions
force	$F$	N	—	Dimensions
density	$ρ$	kg/m³	—	Dimensions
work	$W$	J	—	Dimensions
surface area	$S$	m²	—	Dimensions
height	$h$	m	—	Dimensions
breadth	$b$	m	—	Dimensions
cylinder surface area	$S_{c}$	m²	—	Dimensions
volume of a slab	$V_{s}$	m³	—	Dimensions
relative density	$ρ_{rel}$	—	—	Dimensions
density of water	$ρ_{w}$	kg/m³	—	Dimensions
linear magnification	$m_{l}$	—	—	Dimensions
image length	$l_{i}$	m	—	Dimensions
object length	$l_{o}$	m	—	Dimensions
image area	$A_{i}$	m²	—	Dimensions
object area	$A_{o}$	m²	—	Dimensions
volume of them all	$V_{N}$	m³	—	Dimensions
molar volume	$V_{m}$	m³	—	Dimensions
room to spare	$R_{V}$	—	—	Dimensions

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

V = l^{3}

volume of a cube

v = \frac{l}{t}

average speed

a = \frac{v}{t}

average acceleration

F = m \cdot a

newtons second law

ρ = \frac{m}{V}

mass per volume

W = F \cdot l

force through a distance

S = 6 \cdot l^{2}

surface of a cube

V = \frac{4 \cdot π \cdot r^{3}}{3}

volume of a sphere

S_{c} = 2 \cdot π \cdot r \cdot (r + h)

surface of a cylinder

A = l \cdot b

area of a rectangle

V_{s} = l \cdot b \cdot h

volume from three sides

ρ_{rel} = \frac{ρ}{ρ_{w}}

density against water

m_{l} = \frac{l_{i}}{l_{o}}

lengths magnified

{m_{l}}^{2} = \frac{A_{i}}{A_{o}}

areas magnified

V_{N} = N \cdot V

atoms side by side

R_{V} = \frac{V_{m}}{V_{N}}

molar over atomic volume