Weighing Mars by Phobos

solved EarthSatellites

Q. The planet Mars has two moons $m_{m} = M_{o}$ , Phobos and Deimos. Phobos has a period of 7 hours, 39 minutes $T = 459$ min and an orbital radius of 9.4 × 10³ km $r_{o} = 9.4 \times 10^{3}$ km. What is the mass of Mars, in kg?

Inputs

radius

r_{o}

9400 km

Outputs

mass

m_{m}

How it's solved

Solved with EarthSatellites

Given

$r_{o} = 9.4 \times 10^{3}$ km
$T = 459$ min from “7 hours, 39 minutes”
$m_{m} = M_{o}$ from “two moons”

Find $m_{m}$ .

Working

$r_{o} = 9.4 \times 10^{3}$ km $= 9.4 \times 10^{6}$ m
$T = 459$ min $= 27540$ s
once round a period(round) $ω_{p} = 2 \cdot \frac{π}{T}$
$ω_{p} = 2 \cdot \frac{3.14159265}{27540}$
$ω_{p} = 2.28147615 \times 10^{−4}$ rad/s
speed round the circle(round) $v_{p} = ω_{p} \cdot r_{o}$
$v_{p} = 2.28147615 \times 10^{−4} \cdot 9.4 \times 10^{6}$
$v_{p} = 2144.58758$ m/s
held in orbit by gravity(round) ${v_{p}}^{2} = \frac{G \cdot M_{o}}{r_{o}}$
${2144.58758}^{2} = \frac{6.674 \times 10^{−11} \cdot M_{o}}{9.4 \times 10^{6}}$
$M_{o} = 6.47782518 \times 10^{23}$ kg
two moons $m_{m} = M_{o}$
$m_{m} = 6.47782518 \times 10^{23}$
$m_{m} = 6.478 \times 10^{23}$ kg
$m_{m} \approx 6.48 \times 10^{23}$ kgto 3 significant figures

Conclusion

Mars's pull is what keeps Phobos going round, so how fast it has to go round at that distance tells how strongly Mars pulls, and so how heavy Mars is.

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_mars_by_phobos do4  kind :numerical5  describes "The planet Mars has two moons, Phobos and Deimos. Phobos has a period of 7 hours, 39 minutes and an " \6            "orbital radius of %{radius}. %{mass}, in kg?"7  given radius: "9.4e3 km"8  asks mass: "What is the mass of Mars"9end

solution.rb

1require_relative "../questions/weighing_mars_by_phobos"2require_relative "../science/gravitation/earth_satellites"34solve :weighing_mars_by_phobos, using: EarthSatellites do5  objects :phobos, :mars6  paths :orbit7  phobos.goes(:round, on: orbit)89  take radius: orbit.radius1011  deduce_from "7 hours, 39 minutes", round.period => "459 min"12  deduce_from "two moons", mars.mass => orbit.central_mass1314  mass = mars.mass15  answer mass:, in: "kg", figures: 31617  conclusion "Mars's pull is what keeps Phobos going round, so how fast it has to go round at that " \18             "distance tells how strongly Mars pulls, and so how heavy Mars is."19end

playground.rb

1require_relative "../solutions/weighing_mars_by_phobos"23play_with :weighing_mars_by_phobos do4  title "Weighing Mars by Phobos"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 phobos during round	$Δ x_{p}$	m	—
position of phobos at end of round	$x_{1,p}$	m	—
position of phobos at start of round	$x_{0,p}$	m	—
average velocity of phobos during round	$v_{av,p}$	m/s	—
time taken of round	$Δ t$	s	0 … 1e+05
average speed of phobos during round	$c_{av,p}$	m/s	—
path length of phobos during round	$d_{p}$	m	—
speed of phobos during round	$v_{p}$	m/s	0 … ∞
angular speed of phobos during round	$ω_{p}$	rad/s	—
radius of orbit	$r_{o}$	m	0 … ∞
period of round	$T$	s	0 … ∞
frequency of phobos during round	$ν_{p}$	Hz	0 … ∞
revolutions of phobos during round	$n_{rev,p}$	—	—
centripetal acceleration of phobos during round	$a_{c,p}$	m/s²	—
against gravity of phobos during round	$q_{p}$	—	—
centripetal force on phobos during round	$F_{c,p}$	N	—
mass of phobos	$m_{p}$	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 phobos during round	$E_{b,p}$	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 phobos	$g_{s,p}$	m/s²	—
radius of the body of phobos	$R_{p}$	m	0 … ∞
surface gravity of mars	$g_{s,m}$	m/s²	—
mass of mars	$m_{m}$	kg	0 … ∞
radius of the body of mars	$R_{m}$	m	0 … ∞
gravitational intensity of ground by phobos	$E_{g,gp}$	m/s²	—
separation of ground from phobos	$r_{g,gp}$	m	0 … ∞
gravitational intensity of ground by mars	$E_{g,gm}$	m/s²	—
separation of ground from mars	$r_{g,gm}$	m	0 … ∞
gravitational intensity of phobos by ground	$E_{g,pg}$	m/s²	—
separation of phobos from ground	$r_{g,p}$	m	0 … ∞
gravitational intensity of phobos by mars	$E_{g,pm}$	m/s²	—
separation of phobos from mars	$r_{g,pm}$	m	0 … ∞
gravitational intensity of mars by ground	$E_{g,mg}$	m/s²	—
separation of mars from ground	$r_{g,m}$	m	0 … ∞
gravitational intensity of mars by phobos	$E_{g,mp}$	m/s²	—
separation of mars from phobos	$r_{g,mp}$	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_{p} = x_{1,p} - x_{0,p}

change of position(round)

v_{av,p} = \frac{Δ x_{p}}{Δ t}

displacement a time(round)

c_{av,p} = \frac{d_{p}}{Δ t}

path a time(round)

v_{p} = ω_{p} \cdot r_{o}

speed round the circle(round)

ω_{p} = 2 \cdot \frac{π}{T}

once round a period(round)

ν_{p} = \frac{1}{T}

rounds a second(round)

ν_{p} = \frac{n_{rev,p}}{Δ t}

rounds over the time(round)

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

towards the centre(round)

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

measured in gs(round)

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

mass times the turning(round)

{v_{p}}^{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,p} = \frac{G \cdot M_{o} \cdot m_{p}}{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,p} = \frac{G \cdot m_{p}}{{R_{p}}^{2}}

gravity at the surface(phobos)

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

gravity at the surface(mars)

E_{g,gp} = \frac{G \cdot m_{p}}{{r_{g,gp}}^{2}}

law of gravitation(ground, phobos)

E_{g,gm} = \frac{G \cdot m_{m}}{{r_{g,gm}}^{2}}

law of gravitation(ground, mars)

E_{g,pg} = \frac{G \cdot m_{g}}{{r_{g,p}}^{2}}

law of gravitation(phobos, ground)

E_{g,pm} = \frac{G \cdot m_{m}}{{r_{g,pm}}^{2}}

law of gravitation(phobos, mars)

E_{g,mg} = \frac{G \cdot m_{g}}{{r_{g,m}}^{2}}

law of gravitation(mars, ground)

E_{g,mp} = \frac{G \cdot m_{p}}{{r_{g,mp}}^{2}}

law of gravitation(mars, phobos)

m_{m} = M_{o}

two moons