Weighing the Sun by the Earth

solved EarthSatellites

Q. How will you 'weigh the sun $m_{s} = M_{o}$ ', that is estimate its mass? The mean orbital radius of the earth around the sun $T = 1$ y is 1.5 × 10⁸ km $r_{o} = 1.5 \times 10^{8}$ km. What is the mass of the sun, in kg?

Inputs

radius

r_{o}

1.5e+08 km

Outputs

mass

m_{s}

How it's solved

Solved with EarthSatellites

Given

$r_{o} = 1.5 \times 10^{8}$ km
$T = 1$ y from “the earth around the sun”
$m_{s} = M_{o}$ from “weigh the sun”

Find $m_{s}$ .

Working

$r_{o} = 1.5 \times 10^{8}$ km $= 1.5 \times 10^{11}$ m
$T = 1$ y $= 3.15576 \times 10^{7}$ s
once round a period(year) $ω_{e} = 2 \cdot \frac{π}{T}$
$ω_{e} = 2 \cdot \frac{3.14159265}{3.15576 \times 10^{7}}$
$ω_{e} = 1.99102128 \times 10^{−7}$ rad/s
speed round the circle(year) $v_{e} = ω_{e} \cdot r_{o}$
$v_{e} = 1.99102128 \times 10^{−7} \cdot 1.5 \times 10^{11}$
$v_{e} = 29865.3192$ m/s
held in orbit by gravity(year) ${v_{e}}^{2} = \frac{G \cdot M_{o}}{r_{o}}$
${29865.3192}^{2} = \frac{6.674 \times 10^{−11} \cdot M_{o}}{1.5 \times 10^{11}}$
$M_{o} = 2.00465378 \times 10^{30}$ kg
weigh the sun $m_{s} = M_{o}$
$m_{s} = 2.00465378 \times 10^{30}$
$m_{s} = 2.005 \times 10^{30}$ kg
$m_{s} \approx 2.0 \times 10^{30}$ kgto 2 significant figures

Conclusion

The sun's pull keeps the earth going round once a year, so Kepler's law of periods for the earth's orbit holds the sun's mass.

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 :weighing_the_sun_by_the_earth do4  kind :numerical5  describes "How will you 'weigh the sun', that is estimate its mass? The mean orbital radius of the earth around " \6            "the sun is %{radius}. %{mass}, in kg?"7  given radius: "1.5e8 km"8  asks mass: "What is the mass of the sun"9end

solution.rb

1require_relative "../questions/weighing_the_sun_by_the_earth"2require_relative "../science/gravitation/earth_satellites"34solve :weighing_the_sun_by_the_earth, using: EarthSatellites do5  objects :earth, :sun6  paths :orbit7  earth.goes(:year, on: orbit)89  take radius: orbit.radius1011  deduce_from "the earth around the sun", year.period => "1 y"12  deduce_from "weigh the sun", sun.mass => orbit.central_mass1314  mass = sun.mass15  answer mass:, in: "kg", figures: 21617  conclusion "The sun's pull keeps the earth going round once a year, so Kepler's law of periods for the " \18             "earth's orbit holds the sun's mass."19end

playground.rb

1require_relative "../solutions/weighing_the_sun_by_the_earth"23play_with :weighing_the_sun_by_the_earth do4  title "Weighing the Sun by the Earth"5end

The laws it drew on

quantity	written	units	within	from
pi	$π$	—	0 … ∞	UniformCircularMotion
gravitational constant	$G$	N m² kg⁻²	—	UniversalGravitation
mass	$m$	kg	0 … ∞	UniversalGravitation
radius of the body	$R$	m	0 … ∞	UniversalGravitation
surface gravity	$g_{s}$	m/s²	—	UniversalGravitation
separation	$r_{g}$	m	0 … ∞	UniversalGravitation
gravitational intensity	$E_{g}$	m/s²	—	UniversalGravitation
position	$x$	m	—	Velocity
displacement	$Δ x$	m	—	Velocity
path length	$d$	m	—	Velocity
time taken	$Δ t$	s	0 … 1e+05	Velocity
average velocity	$v_{av}$	m/s	—	Velocity
average speed	$c_{av}$	m/s	—	Velocity
starting position	$x_{0}$	m	—	Velocity
final position	$x$	m	—	Velocity
radius	$r$	m	0 … ∞	UniformCircularMotion
speed	$v$	m/s	0 … ∞	UniformCircularMotion
angular speed	$ω$	rad/s	—	UniformCircularMotion
revolutions	$n_{rev}$	—	—	UniformCircularMotion
frequency	$ν$	Hz	0 … ∞	UniformCircularMotion
centripetal acceleration	$a_{c}$	m/s²	—	UniformCircularMotion
against gravity	$q$	—	—	UniformCircularMotion
period	$T$	s	0 … ∞	UniformCircularMotion
gravity	$g$	m/s²	—	UniformCircularMotion
centripetal force	$F_{c}$	N	—	UniformCircularMotion
central mass	$M$	kg	—	EarthSatellites
central radius	$R_{c}$	m	—	EarthSatellites
height of the orbit	$h$	m	—	EarthSatellites
keplers constant	$k_{K}$	s²/m³	—	EarthSatellites
binding energy	$E_{b}$	J	—	EarthSatellites
displacement of earth during year	$Δ x_{e}$	m	—
position of earth at end of year	$x_{1,e}$	m	—
position of earth at start of year	$x_{0,e}$	m	—
average velocity of earth during year	$v_{av,e}$	m/s	—
time taken of year	$Δ t$	s	0 … 1e+05
average speed of earth during year	$c_{av,e}$	m/s	—
path length of earth during year	$d_{e}$	m	—
speed of earth during year	$v_{e}$	m/s	0 … ∞
angular speed of earth during year	$ω_{e}$	rad/s	—
radius of orbit	$r_{o}$	m	0 … ∞
period of year	$T$	s	0 … ∞
frequency of earth during year	$ν_{e}$	Hz	0 … ∞
revolutions of earth during year	$n_{rev,e}$	—	—
centripetal acceleration of earth during year	$a_{c,e}$	m/s²	—
against gravity of earth during year	$q_{e}$	—	—
centripetal force on earth during year	$F_{c,e}$	N	—
mass of earth	$m_{e}$	kg	0 … ∞
central mass of orbit	$M_{o}$	kg	—
keplers constant of orbit	$k_{K,o}$	s²/m³	—
central radius of orbit	$R_{c,o}$	m	—
height of the orbit of orbit	$h_{o}$	m	—
binding energy of earth during year	$E_{b,e}$	J	—
surface gravity of ground	$g_{s,g}$	m/s²	—
mass of ground	$m_{g}$	kg	0 … ∞
radius of the body of ground	$R_{g}$	m	0 … ∞
surface gravity of earth	$g_{s,e}$	m/s²	—
radius of the body of earth	$R_{e}$	m	0 … ∞
surface gravity of sun	$g_{s,s}$	m/s²	—
mass of sun	$m_{s}$	kg	0 … ∞
radius of the body of sun	$R_{s}$	m	0 … ∞
gravitational intensity of ground by earth	$E_{g,ge}$	m/s²	—
separation of ground from earth	$r_{g,ge}$	m	0 … ∞
gravitational intensity of ground by sun	$E_{g,gs}$	m/s²	—
separation of ground from sun	$r_{g,gs}$	m	0 … ∞
gravitational intensity of earth by ground	$E_{g,eg}$	m/s²	—
separation of earth from ground	$r_{g,e}$	m	0 … ∞
gravitational intensity of earth by sun	$E_{g,es}$	m/s²	—
separation of earth from sun	$r_{g,es}$	m	0 … ∞
gravitational intensity of sun by ground	$E_{g,sg}$	m/s²	—
separation of sun from ground	$r_{g,s}$	m	0 … ∞
gravitational intensity of sun by earth	$E_{g,se}$	m/s²	—
separation of sun from earth	$r_{g,se}$	m	0 … ∞

x_{0} \equiv x_{i} (start of motion)

starting position

x \equiv x_{i} (end of motion)

final position

g_{s,b} = \frac{G \cdot m_{b}}{{R_{b}}^{2}}

gravity at the surface

E_{g,oo} = \frac{G \cdot m_{o}}{{r_{g,oo}}^{2}}

law of gravitation

d_{i} (whole) = d_{i} (part 1) + d_{i} (part 2)

path in parts

Δ t_{w} = Δ t_{p} + Δ t_{p}

time in parts

Δ x_{e} = x_{1,e} - x_{0,e}

change of position(year)

v_{av,e} = \frac{Δ x_{e}}{Δ t}

displacement a time(year)

c_{av,e} = \frac{d_{e}}{Δ t}

path a time(year)

v_{e} = ω_{e} \cdot r_{o}

speed round the circle(year)

ω_{e} = 2 \cdot \frac{π}{T}

once round a period(year)

ν_{e} = \frac{1}{T}

rounds a second(year)

ν_{e} = \frac{n_{rev,e}}{Δ t}

rounds over the time(year)

a_{c,e} = \frac{{v_{e}}^{2}}{r_{o}}

towards the centre(year)

q_{e} = \frac{a_{c,e}}{g}

measured in gs(year)

F_{c,e} = m_{e} \cdot a_{c,e}

mass times the turning(year)

{v_{e}}^{2} = \frac{G \cdot M_{o}}{r_{o}}

held in orbit by gravity(year)

k_{K,o} = \frac{4 \cdot π^{2}}{G \cdot M_{o}}

keplers constant of an orbit(year)

T^{2} = k_{K,o} \cdot {r_{o}}^{3}

law of periods(year)

r_{o} = R_{c,o} + h_{o}

orbit radius from the centre(year)

E_{b,e} = \frac{G \cdot M_{o} \cdot m_{e}}{2 \cdot r_{o}}

binding energy of an orbit(year)

g_{s,g} = \frac{G \cdot m_{g}}{{R_{g}}^{2}}

gravity at the surface(ground)

g_{s,e} = \frac{G \cdot m_{e}}{{R_{e}}^{2}}

gravity at the surface(earth)

g_{s,s} = \frac{G \cdot m_{s}}{{R_{s}}^{2}}

gravity at the surface(sun)

E_{g,ge} = \frac{G \cdot m_{e}}{{r_{g,ge}}^{2}}

law of gravitation(ground, earth)

E_{g,gs} = \frac{G \cdot m_{s}}{{r_{g,gs}}^{2}}

law of gravitation(ground, sun)

E_{g,eg} = \frac{G \cdot m_{g}}{{r_{g,e}}^{2}}

law of gravitation(earth, ground)

E_{g,es} = \frac{G \cdot m_{s}}{{r_{g,es}}^{2}}

law of gravitation(earth, sun)

E_{g,sg} = \frac{G \cdot m_{g}}{{r_{g,s}}^{2}}

law of gravitation(sun, ground)

E_{g,se} = \frac{G \cdot m_{e}}{{r_{g,se}}^{2}}

law of gravitation(sun, earth)

m_{s} = M_{o}

weigh the sun