Weighing Jupiter by Io

Q. Io, one of the satellites of Jupiter, has an orbital period of 1.769 d and the radius of the orbit is 4.22 × 10⁸ m. What is the mass of Jupiter, in kg? expects 1.9 × 10²⁷ kg

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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?

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Inputs

period

T

1.769 d

radius

r_{o}

422000000.00 m

Outputs

mass

m_{j}

The solution

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 Ruby

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

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

1require_relative "../solutions/weighing_jupiter_by_io"23play_with :weighing_jupiter_by_io do4  title "Weighing Jupiter by Io"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 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