Good morning, Interweb.
Let's worldbuild.
Today, let's embark on bit of terrestrial planet moonbuilding.
For the purposes of this build, I'm going to classify moons by size and composition.
Size:
All natural satellites can either be major moons or minor moons.
Minor moons are small satellites that lack the necessary gravity to force them into a
spherical shape.
These potato shaped bodies, like the moons of Mars, will be no greater than about 200-300km
in size.
Major moons, on the other hand, like our own Moon, are gravitationally rounded, as such
they must have radii larger than 200-300 km.
Composition:
Moons can be either predominantly rocky, again like our Moon, or predominantly icy like Mimas.
A grid like this, although simplistic, is very useful tool when it comes to moonbuilding.
So, here's a system comprised solely of terrestrial planets – three inner worlds (one in the
habitable zone) and one outer planet.
Before we plot the location of any moons it's worth going over some general guidelines.
1) Amount!
For the most part, terrestrial planets will have very few satellites if any They may occasionally
capture a few asteroid minor moons but are unlikely to have major moons.
2) Habitability!
Terrestrial planets intended to be habitable must have at least one major moon.
Major moons are pretty much a prerequisite for habitability – they help moderate a
planets climate by stabilizing it's axial tilt.
Without a moon, Earth would look less like this and more like this.
3) Distance!
Close in planets will have less moons than distant planets.
Compare moonless Mercury with Pluto's veritable orgy of moons.
Aaaand...
4) Composition!
Inner system moons will be predominantly rocky and outer system moons will be predominantly
icy.
So with all this in mind, I'll elect to give this guy no moons, this guy one captured asteroid
minor moon.
My habitable world must get a rocky major moon (which probably formed as result of a
collision similar to that which formed our moon) and I'll less give this distant guy...oh,
i dunno...4 icy minor moons.
Just 'cause I can.
Alrighty then, Moon-building engage!
As the habitable world will likely be the most important locale in the system, let's
begin by constructing it's moon.
First up, let's figure out what it's made of.
We know it needs to be predominantly rocky so let's say it's comprised of 90% silicate
rock, 9% iron and 1% water ice.
Simply multiply the density of each material by the percentage you chose to get the overall
density of your moon.
In this case we're looking at a density of about 3.62 g/cm^3 aka 0.66 times the density
of Earth.
Next we can use this equation to find our moon's mass and radius: M, the mass of our
moon in earth masses is equal to it's radius, in earth radii, cubed, multiplied by it's
density, in earth densities – which we already have.
Because we're building a major moon, we want to selected a radius greater than 200-300km
aka 0.03 to about 0.05 Earth radii.
Let's say our moon will be just under a third the size of earth which gives us a mass of
0.018 earth masses.
This makes it slightly more massive than Earth's moon.
Which is perfect; in order to ensure that our moon will stabilize our habitable world
we need to design it in such a way so as to be physically comparable to Earth's moon.
As a quick aside, a satellites mass must always be less than or equal to the mass of the planet
it orbits.
However, the closer the two objects are in mass the more the system begins to resemble
a double planet system ala Pluto and Charon.
A useful point for those wishing to construct double habitable planets.
Now, let's plug our mass and radius into this equation to get our moons surface gravity
relative to Earths.
In this case, it's surface gravity is 0.2 – about 23% higher than that of Earth's
moon.
So, that's the essential physical parameters take care of...let's put the moon into orbit.
To do this we're going to need the know the physical characteristics of our star and our
planet.
I've covered the construction of stars and planets in previous videos so click the links
on screen, or in the doobly-doo, if you need a bit of a refresher.
Tl;dr my star here is identical to our sun and the planet is a slightly larger version
of earth.
Anyways!
Planets, like all astronomical objects, are encased in a sort of gravitational bubble
we call a Hill Sphere.
Any object inside a planet's Hill Sphere will be gravitationally bound to that planet.
So, by definition, satellites must be placed inside this bubble the outer limit of which
is given by the equation: “a” the semi major axis of the planets orbit in AU, times
the cubed root of the mass of the planet in Earth masses, divided by the mass of the star
in solar masses multiplied 235.
Plugging the relevant numbers in gets me a value of roughly 269.
That is, this planets Hill sphere extends out roughly 269 earth radii from the centre
of the planet.
Those who watched my “rings” video, will know that a moon cannot orbit arbitrarily
close to a planet so the inner limit here is given by the Roche limit: 2.44 times the
radius of our planet in Earth radii, times the density of the planet divided by the density
of the moon to the 1/3 power.
Again, whack in the relevant numbers and out pops 2.65.
So, the inner limit lies 2.65 earth radii from the centre of our planet.
Bit of bonus info here is that moons also have Hill Spheres and Roche limits meaning
it's theoretically possible for to construct moons of moons.
Regardless...this right here is moon territory!
In general, terrestrial planet moons will be what's known as regular satellites.
That is, they will orbit relatively close to the planet on prograde, uninclined, nearly
circular orbits.
“Close to the planet” generally means within 1/2 of the maximum value here.
So, I'll put my moon into orbit about 80 earth radii out.
I'll give it a prograde orbit, a very low orbital inclination, say 2 degrees, and very
low eccentricity - about 0.01.
Oh, and for those wishing to construct two separate major moons ensure that they never
come within 10 planetary radii of each other.
Now, our moons orbital period, which will play an important role in future calendar
building, will be given by: 0.0588 times the square root of the radius of the moon's orbit
in earth radii cubed, divided by the sum of the masses of the planet and the moon in earth
masses.
After another bout of intense number vomit we get an orbital period of 34.15 earth days.
So, lunar months on this world will be roughly 34 earth days long.
And that right there is how one goes about constructing moons for habitable earth-like
planets!
Now...on to the other terrestrial planets!
Uninhabitable terrestrial planets require a very different moon building technique.
Why?
Because...
1) Terrestrial planets shouldn't really be given any major moons.
Earth is anomalous in this regard.
Instead we have to deal with minor moons.
Minor moons aren't spheres so there's less we can/need say about them.
And 2) The stabilization effect of minor moons
is minimal so we can be more liberal when choosing their orbits.
Let's start with this guy.
It's rocky so, by definition, it's density should be in the ball park of 3 g/cm^3 – the
higher the value here to more iron content the minor moon will have.
Useful for those wishing to construct potential mining locations.
I'll go with a density of 4 g/cm^3 aka about 3/4s the density of earth.
The moon is non spherical so it's shape can be whatever we wan't it to be but, for ease
of construction, I'd advocate modeling minor moons as ellipsoids.
Which means we're gonna need to specify three axes: a, b and c.
Let's say this guy is very roughly 80 km x 50km x 30km.
Like this!
'Cause, ya know, who doesn't like fibonacci inspired minor moons.
FYI ellipsoids come in three distinct flavors: tri-axial ellipsoids (where a is greater than
b which in turn is than c), oblate spheroids (where a and b are equal and greater than
c) and prolate spheroids (where a and b are equal but less than c).
Choose at will.
With density, size and shape done, let's talk mass.
The mass of an ellipsoid is given by P the density times 4/3 pi, times the three axes
a,b and c.
To make sure the units work out we have to multiply each of our inputs by 1000.
In this case my moon ends up with a mass of about 2x10^18 kg which seems like a lot but
really it's just 0.00000033 Earth masses.
Now, we could try and set surface gravity but given the irregularity of minor minors,
even I think that'd be overkill.
I mean determining their mass is already sorta overkill...surface gravity would be a total
fatality.
Simply knowing a minor moons composition, size and shape is more than enough really.
That said, let's put this guy into orbit.
Same method applies as before.
Find the Hill sphere, like so, and the Roche limit, like so, and place your moon within
this zone.
Because the minor moons of terrestrial planets will likely be captured asteroids, they can
pretty much orbit anywhere in the zone no restrictions.
So, I'll place this guy at about 30 earth radii out.
Giving him a orbital period of just over 11 earth days.
Minor moon done!
Same stick with our distant terrestrial planet.
I selected 4 densities that indicate an icy composition, i.e., in and around the 1-2 g/cm^3
mark.
The higher density values here indicate a mixture of rock and ice – dirty snowballs
if you will.
I made 3 of them oblate spheroids and one a tri-axial ellipsoid.
And placed each of them in the moon zone whilst ensuring that no two moons came within 10
planetary radii of each other.
And there you have it one terrestrial planetary system all decked out in moons.
Next time, we'll have a look at gas giant moon systems AND the possibility of habitable
moons!!
Stay tuned!