How Far Saturn Is

solved EarthSatellites

Q. A saturn year is 29.5 times $T_{s} = 29.5 \cdot T_{e}$ the earth year $T_{e} = 1$ y. How far is the saturn from the sun if the earth is 1.50 × 10⁸ km $r_{e} = 1.50 \times 10^{8}$ km away from the sun $M_{s} = M_{e}$ ? Answer in km.

Inputs

earth

r_{e}

1.5e+08 km

Outputs

distance

r_{s}

How it's solved

Solved with EarthSatellites

Given

$r_{e} = 1.50 \times 10^{8}$ km
$T_{e} = 1$ y from “the earth year”
$T_{s} = 29.5 \cdot T_{e}$ from “29.5 times”
$M_{s} = M_{e}$ from “away from the sun”

Find $r_{s}$ .

Working

$r_{e} = 1.50 \times 10^{8}$ km $= 1.5 \times 10^{11}$ m
$T_{e} = 1$ y $= 3.15576 \times 10^{7}$ s
29.5 times $T_{s} = 29.5 \cdot T_{e}$
$T_{s} = 29.5 \cdot 3.15576 \times 10^{7}$
$T_{s} = 9.309492 \times 10^{8}$ s
once round a period(earth year) $ω_{E} (e) = 2 \cdot \frac{π}{T_{e}}$
$ω_{E} (e) = 2 \cdot \frac{3.14159265}{3.15576 \times 10^{7}}$
$ω_{E} (e) = 1.99102128 \times 10^{−7}$ rad/s
speed round the circle(earth year) $v_{E} (e) = ω_{E} (e) \cdot r_{e}$
$v_{E} (e) = 1.99102128 \times 10^{−7} \cdot 1.5 \times 10^{11}$
$v_{E} (e) = 29865.3192$ m/s
held in orbit by gravity(earth year) ${v_{E} (e)}^{2} = \frac{G \cdot M_{e}}{r_{e}}$
${29865.3192}^{2} = \frac{6.674 \times 10^{−11} \cdot M_{e}}{1.5 \times 10^{11}}$
$M_{e} = 2.00465378 \times 10^{30}$ kg
away from the sun $M_{s} = M_{e}$
$M_{s} = 2.00465378 \times 10^{30}$
$M_{s} = 2.00465378 \times 10^{30}$ kg
keplers constant of an orbit(saturn year) $k_{K,s} = \frac{4 \cdot π^{2}}{G \cdot M_{s}}$
$k_{K,s} = \frac{4 \cdot {3.14159265}^{2}}{6.674 \times 10^{−11} \cdot 2.00465378 \times 10^{30}}$
$k_{K,s} = 2.95076183 \times 10^{−19}$ s²/m³
law of periods(saturn year) ${T_{s}}^{2} = k_{K,s} \cdot {r_{s}}^{3}$
${(9.309492 \times 10^{8})}^{2} = 2.95076183 \times 10^{−19} \cdot {r_{s}}^{3}$
$r_{s} = 1.432 \times 10^{9}$ km
$r_{s} = 1.43209755 \times 10^{12}$ m $= 1.43209755 \times 10^{9}$ km
$r_{s} \approx 1.43 \times 10^{9}$ kmto 3 significant figures

Conclusion

Round the same sun the square of the period goes as the cube of the orbit's size, so a much longer year means a much wider orbit, though not as many times wider.

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 :how_far_saturn_is do4  kind :numerical5  describes "A saturn year is 29.5 times the earth year. %{distance} if the earth is %{earth} away from the sun? " \6            "Answer in km."7  given earth: "1.50e8 km"8  asks distance: "How far is the saturn from the sun"9end

solution.rb

1require_relative "../questions/how_far_saturn_is"2require_relative "../science/gravitation/earth_satellites"34solve :how_far_saturn_is, using: EarthSatellites do5  objects saturn: :S, earth: :E6  paths :saturn_orbit, :earth_orbit7  saturn.goes(:saturn_year, on: saturn_orbit)8  earth.goes(:earth_year, on: earth_orbit)910  take earth: earth_orbit.radius1112  deduce_from "29.5 times", saturn_year.period => 29.5 * earth_year.period13  deduce_from "away from the sun", saturn_orbit.central_mass => earth_orbit.central_mass14  deduce_from "the earth year", earth_year.period => "1 y"1516  distance = saturn_orbit.radius17  answer distance:, in: "km", figures: 31819  conclusion "Round the same sun the square of the period goes as the cube of the orbit's size, so a much " \20             "longer year means a much wider orbit, though not as many times wider."21end

playground.rb

1require_relative "../solutions/how_far_saturn_is"23play_with :how_far_saturn_is do4  title "How Far Saturn Is"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 saturn during saturn year	$Δ x_{S} (s)$	m	—
position of saturn at end of saturn year	$x_{S} (s₁)$	m	—
position of saturn at start of saturn year	$x_{S} (s₀)$	m	—
displacement of earth during earth year	$Δ x_{E} (e)$	m	—
position of earth at end of earth year	$x_{E} (e₁)$	m	—
position of earth at start of earth year	$x_{E} (e₀)$	m	—
average velocity of saturn during saturn year	$v_{av,S} (s)$	m/s	—
time taken of saturn year	$Δ t_{s}$	s	0 … 1e+05
average velocity of earth during earth year	$v_{av,E} (e)$	m/s	—
time taken of earth year	$Δ t_{e}$	s	0 … 1e+05
average speed of saturn during saturn year	$c_{av,S} (s)$	m/s	—
path length of saturn during saturn year	$d_{S} (s)$	m	—
average speed of earth during earth year	$c_{av,E} (e)$	m/s	—
path length of earth during earth year	$d_{E} (e)$	m	—
speed of saturn during saturn year	$v_{S} (s)$	m/s	0 … ∞
angular speed of saturn during saturn year	$ω_{S} (s)$	rad/s	—
radius of saturn orbit	$r_{s}$	m	0 … ∞
speed of earth during earth year	$v_{E} (e)$	m/s	0 … ∞
angular speed of earth during earth year	$ω_{E} (e)$	rad/s	—
radius of earth orbit	$r_{e}$	m	0 … ∞
period of saturn year	$T_{s}$	s	0 … ∞
period of earth year	$T_{e}$	s	0 … ∞
frequency of saturn during saturn year	$ν_{S} (s)$	Hz	0 … ∞
frequency of earth during earth year	$ν_{E} (e)$	Hz	0 … ∞
revolutions of saturn during saturn year	$n_{rev,S} (s)$	—	—
revolutions of earth during earth year	$n_{rev,E} (e)$	—	—
centripetal acceleration of saturn during saturn year	$a_{c,S} (s)$	m/s²	—
centripetal acceleration of earth during earth year	$a_{c,E} (e)$	m/s²	—
against gravity of saturn during saturn year	$q_{S} (s)$	—	—
against gravity of earth during earth year	$q_{E} (e)$	—	—
centripetal force on saturn during saturn year	$F_{c,S} (s)$	N	—
mass of saturn	$m_{S}$	kg	0 … ∞
centripetal force on earth during earth year	$F_{c,E} (e)$	N	—
mass of earth	$m_{E}$	kg	0 … ∞
central mass of saturn orbit	$M_{s}$	kg	—
central mass of earth orbit	$M_{e}$	kg	—
keplers constant of saturn orbit	$k_{K,s}$	s²/m³	—
keplers constant of earth orbit	$k_{K,e}$	s²/m³	—
central radius of saturn orbit	$R_{c,s}$	m	—
height of the orbit of saturn orbit	$h_{s}$	m	—
central radius of earth orbit	$R_{c,e}$	m	—
height of the orbit of earth orbit	$h_{e}$	m	—
binding energy of saturn during saturn year	$E_{b,S} (s)$	J	—
binding energy of earth during earth year	$E_{b,E} (e)$	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 saturn	$g_{s,S}$	m/s²	—
radius of the body of saturn	$R_{S}$	m	0 … ∞
surface gravity of earth	$g_{s,E}$	m/s²	—
radius of the body of earth	$R_{E}$	m	0 … ∞
gravitational intensity of ground by saturn	$E_{g,gS}$	m/s²	—
separation of ground from saturn	$r_{g,gS}$	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 saturn by ground	$E_{g,Sg}$	m/s²	—
separation of saturn from ground	$r_{g,S}$	m	0 … ∞
gravitational intensity of saturn by earth	$E_{g,SE}$	m/s²	—
separation of saturn from earth	$r_{g,SE}$	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 saturn	$E_{g,ES}$	m/s²	—
separation of earth from saturn	$r_{g,ES}$	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_{S} (s) = x_{S} (s₁) - x_{S} (s₀)

change of position(saturn year)

Δ x_{E} (e) = x_{E} (e₁) - x_{E} (e₀)

change of position(earth year)

v_{av,S} (s) = \frac{Δ x_{S} (s)}{Δ t_{s}}

displacement a time(saturn year)

v_{av,E} (e) = \frac{Δ x_{E} (e)}{Δ t_{e}}

displacement a time(earth year)

c_{av,S} (s) = \frac{d_{S} (s)}{Δ t_{s}}

path a time(saturn year)

c_{av,E} (e) = \frac{d_{E} (e)}{Δ t_{e}}

path a time(earth year)

v_{S} (s) = ω_{S} (s) \cdot r_{s}

speed round the circle(saturn year)

v_{E} (e) = ω_{E} (e) \cdot r_{e}

speed round the circle(earth year)

ω_{S} (s) = 2 \cdot \frac{π}{T_{s}}

once round a period(saturn year)

ω_{E} (e) = 2 \cdot \frac{π}{T_{e}}

once round a period(earth year)

ν_{S} (s) = \frac{1}{T_{s}}

rounds a second(saturn year)

ν_{E} (e) = \frac{1}{T_{e}}

rounds a second(earth year)

ν_{S} (s) = \frac{n_{rev,S} (s)}{Δ t_{s}}

rounds over the time(saturn year)

ν_{E} (e) = \frac{n_{rev,E} (e)}{Δ t_{e}}

rounds over the time(earth year)

a_{c,S} (s) = \frac{{v_{S} (s)}^{2}}{r_{s}}

towards the centre(saturn year)

a_{c,E} (e) = \frac{{v_{E} (e)}^{2}}{r_{e}}

towards the centre(earth year)

q_{S} (s) = \frac{a_{c,S} (s)}{g}

measured in gs(saturn year)

q_{E} (e) = \frac{a_{c,E} (e)}{g}

measured in gs(earth year)

F_{c,S} (s) = m_{S} \cdot a_{c,S} (s)

mass times the turning(saturn year)

F_{c,E} (e) = m_{E} \cdot a_{c,E} (e)

mass times the turning(earth year)

{v_{S} (s)}^{2} = \frac{G \cdot M_{s}}{r_{s}}

held in orbit by gravity(saturn year)

{v_{E} (e)}^{2} = \frac{G \cdot M_{e}}{r_{e}}

held in orbit by gravity(earth year)

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

keplers constant of an orbit(saturn year)

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

keplers constant of an orbit(earth year)

{T_{s}}^{2} = k_{K,s} \cdot {r_{s}}^{3}

law of periods(saturn year)

{T_{e}}^{2} = k_{K,e} \cdot {r_{e}}^{3}

law of periods(earth year)

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

orbit radius from the centre(saturn year)

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

orbit radius from the centre(earth year)

E_{b,S} (s) = \frac{G \cdot M_{s} \cdot m_{S}}{2 \cdot r_{s}}

binding energy of an orbit(saturn year)

E_{b,E} (e) = \frac{G \cdot M_{e} \cdot m_{E}}{2 \cdot r_{e}}

binding energy of an orbit(earth year)

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

gravity at the surface(ground)

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

gravity at the surface(saturn)

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

gravity at the surface(earth)

E_{g,gS} = \frac{G \cdot m_{S}}{{r_{g,gS}}^{2}}

law of gravitation(ground, saturn)

E_{g,gE} = \frac{G \cdot m_{E}}{{r_{g,gE}}^{2}}

law of gravitation(ground, earth)

E_{g,Sg} = \frac{G \cdot m_{g}}{{r_{g,S}}^{2}}

law of gravitation(saturn, ground)

E_{g,SE} = \frac{G \cdot m_{E}}{{r_{g,SE}}^{2}}

law of gravitation(saturn, earth)

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, saturn)

T_{s} = 29.5 \cdot T_{e}

29.5 times

M_{s} = M_{e}

away from the sun