How Far Saturn Is

Q. A saturn year is 29.5 times the earth year. How far is the saturn from the sun if the earth is 1.50 × 10⁸ km away from the sun? Answer in km. expects 1.43 × 10⁹ km

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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.

The playground

Inputs

earth

r_{e}

1.5e+08 km

Outputs

distance

r_{s}

The solution

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 Ruby

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

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

1require_relative "../solutions/how_far_saturn_is"23play_with :how_far_saturn_is do4  title "How Far Saturn Is"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 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