Weighing the Sun by the Earth

Q. How will you 'weigh the sun', that is estimate its mass? The mean orbital radius of the earth around the sun is 1.5 × 10⁸ km. What is the mass of the sun, in kg? expects 2.0 × 10³⁰ kg

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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?

The playground

Inputs

radius

r_{o}

1.5e+08 km

Outputs

mass

m_{s}

The solution

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 Ruby

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

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

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

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