Weighing Jupiter by Io

solved EarthSatellites

Q. Io, one of the satellites of Jupiter $m_{j} = M_{o}$ , has an orbital period of 1.769 d $T = 1.769$ d and the radius of the orbit is 4.22 × 10⁸ m $r_{o} = 4.22 \times 10^{8}$ m. What is the mass of Jupiter, in kg?

Inputs

period

T

1.769 d

radius

r_{o}

422000000.00 m

Outputs

mass

m_{j}

How it's solved

Solved with EarthSatellites

Given

$T = 1.769$ d
$r_{o} = 4.22 \times 10^{8}$ m
$m_{j} = M_{o}$ from “satellites of Jupiter”

Find $m_{j}$ .

Working

$T = 1.769$ d $= 152841.6$ s
once round a period(round) $ω_{i} = 2 \cdot \frac{π}{T}$
$ω_{i} = 2 \cdot \frac{3.14159265}{152841.6}$
$ω_{i} = 4.11091307 \times 10^{−5}$ rad/s
speed round the circle(round) $v_{i} = ω_{i} \cdot r_{o}$
$v_{i} = 4.11091307 \times 10^{−5} \cdot 4.22 \times 10^{8}$
$v_{i} = 17348.0531$ m/s
held in orbit by gravity(round) ${v_{i}}^{2} = \frac{G \cdot M_{o}}{r_{o}}$
${17348.0531}^{2} = \frac{6.674 \times 10^{−11} \cdot M_{o}}{4.22 \times 10^{8}}$
$M_{o} = 1.90295157 \times 10^{27}$ kg
satellites of Jupiter $m_{j} = M_{o}$
$m_{j} = 1.90295157 \times 10^{27}$
$m_{j} = 1.903 \times 10^{27}$ kg
$m_{j} \approx 1.9 \times 10^{27}$ kgto 2 significant figures

Conclusion

Jupiter's pull keeps Io going round, so Kepler's law of periods for Io's orbit holds Jupiter's mass, which comes out about a thousandth of the Sun's.

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_jupiter_by_io do4  kind :numerical5  describes "Io, one of the satellites of Jupiter, has an orbital period of %{period} and the radius of the orbit " \6            "is %{radius}. %{mass}, in kg?"7  given period: "1.769 d"8  given radius: "4.22e8 m"9  asks mass: "What is the mass of Jupiter"10end

solution.rb

1require_relative "../questions/weighing_jupiter_by_io"2require_relative "../science/gravitation/earth_satellites"34solve :weighing_jupiter_by_io, using: EarthSatellites do5  objects :io, :jupiter6  paths :orbit7  io.goes(:round, on: orbit)89  take period: round.period10  take radius: orbit.radius1112  deduce_from "satellites of Jupiter", jupiter.mass => orbit.central_mass1314  mass = jupiter.mass15  answer mass:, in: "kg", figures: 21617  conclusion "Jupiter's pull keeps Io going round, so Kepler's law of periods for Io's orbit holds Jupiter's " \18             "mass, which comes out about a thousandth of the Sun's."19end

playground.rb

1require_relative "../solutions/weighing_jupiter_by_io"23play_with :weighing_jupiter_by_io do4  title "Weighing Jupiter by Io"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 io during round	$Δ x_{i}$	m	—
position of io at end of round	$x_{1,i}$	m	—
position of io at start of round	$x_{0,i}$	m	—
average velocity of io during round	$v_{av,i}$	m/s	—
time taken of round	$Δ t$	s	0 … 1e+05
average speed of io during round	$c_{av,i}$	m/s	—
path length of io during round	$d_{i}$	m	—
speed of io during round	$v_{i}$	m/s	0 … ∞
angular speed of io during round	$ω_{i}$	rad/s	—
radius of orbit	$r_{o}$	m	0 … ∞
period of round	$T$	s	0 … ∞
frequency of io during round	$ν_{i}$	Hz	0 … ∞
revolutions of io during round	$n_{rev,i}$	—	—
centripetal acceleration of io during round	$a_{c,i}$	m/s²	—
against gravity of io during round	$q_{i}$	—	—
centripetal force on io during round	$F_{c,i}$	N	—
mass of io	$m_{i}$	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 io during round	$E_{b,i}$	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 io	$g_{s,i}$	m/s²	—
radius of the body of io	$R_{i}$	m	0 … ∞
surface gravity of jupiter	$g_{s,j}$	m/s²	—
mass of jupiter	$m_{j}$	kg	0 … ∞
radius of the body of jupiter	$R_{j}$	m	0 … ∞
gravitational intensity of ground by io	$E_{g,gi}$	m/s²	—
separation of ground from io	$r_{g,gi}$	m	0 … ∞
gravitational intensity of ground by jupiter	$E_{g,gj}$	m/s²	—
separation of ground from jupiter	$r_{g,gj}$	m	0 … ∞
gravitational intensity of io by ground	$E_{g,ig}$	m/s²	—
separation of io from ground	$r_{g,i}$	m	0 … ∞
gravitational intensity of io by jupiter	$E_{g,ij}$	m/s²	—
separation of io from jupiter	$r_{g,ij}$	m	0 … ∞
gravitational intensity of jupiter by ground	$E_{g,jg}$	m/s²	—
separation of jupiter from ground	$r_{g,j}$	m	0 … ∞
gravitational intensity of jupiter by io	$E_{g,ji}$	m/s²	—
separation of jupiter from io	$r_{g,ji}$	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_{i} = x_{1,i} - x_{0,i}

change of position(round)

v_{av,i} = \frac{Δ x_{i}}{Δ t}

displacement a time(round)

c_{av,i} = \frac{d_{i}}{Δ t}

path a time(round)

v_{i} = ω_{i} \cdot r_{o}

speed round the circle(round)

ω_{i} = 2 \cdot \frac{π}{T}

once round a period(round)

ν_{i} = \frac{1}{T}

rounds a second(round)

ν_{i} = \frac{n_{rev,i}}{Δ t}

rounds over the time(round)

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

towards the centre(round)

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

measured in gs(round)

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

mass times the turning(round)

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

held in orbit by gravity(round)

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

keplers constant of an orbit(round)

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

law of periods(round)

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

orbit radius from the centre(round)

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

binding energy of an orbit(round)

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

gravity at the surface(ground)

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

gravity at the surface(io)

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

gravity at the surface(jupiter)

E_{g,gi} = \frac{G \cdot m_{i}}{{r_{g,gi}}^{2}}

law of gravitation(ground, io)

E_{g,gj} = \frac{G \cdot m_{j}}{{r_{g,gj}}^{2}}

law of gravitation(ground, jupiter)

E_{g,ig} = \frac{G \cdot m_{g}}{{r_{g,i}}^{2}}

law of gravitation(io, ground)

E_{g,ij} = \frac{G \cdot m_{j}}{{r_{g,ij}}^{2}}

law of gravitation(io, jupiter)

E_{g,jg} = \frac{G \cdot m_{g}}{{r_{g,j}}^{2}}

law of gravitation(jupiter, ground)

E_{g,ji} = \frac{G \cdot m_{i}}{{r_{g,ji}}^{2}}

law of gravitation(jupiter, io)

m_{j} = M_{o}

satellites of Jupiter