Paradigm, the Planet : How It’s Done

In the pre­vi­ous post I men­tioned how I want­ed to “set the stage” by build­ing inwards, start­ing super-macro lev­el with con­struct­ing a solar sys­tem and plan­et. Then I could place things on my the­o­ret­i­cal globe and run with it.

What I failed to men­tion is how dif­fi­cult this work has been. In pur­suit of real­ism I’ve tried to con­struct the world using actu­al (astro)physics and math. And I’ve been putting off this post in par­tic­u­lar. A year or so ago I had suf­fered through all of the math and con­struc­tion, and failed. I’ll explain below what hap­pened, but I end­ed up putting the notes aside and mov­ing to work on some­thing else. Need­less to say the break from this top­ic was much needed.

When I returned to the top­ic of space-physics (writ­ing this post) I real­ized I had lost my notes. So I had to build con­tent on a top­ic I have par­tic­u­lar dif­fi­cul­ty with, and I didn’t have any of my pre­vi­ous work to help me. And if you’ve ever had to rewrite a lost paper or start work over after los­ing pre­vi­ous drafts, you’ll know the feel­ing of annoy­ance that the “new” work you’re cre­at­ing is close to what you wrote first, but frus­trat­ing­ly not the same.

In prepar­ing for this post, I found my notes. And there was much rejoic­ing. Below I’ll still do my best to explain my method­ol­o­gy, but cer­tain num­bers I choose may seem pulled out of thin air; assume I’ve cho­sen those num­bers using real work references. 

I sat down to con­tin­ue my work in the pre­vi­ous post (The Solar Sys­tem) and fin­ish up orga­niz­ing Paradigm’s cos­mic neigh­bors. After find­ing my lost notes, I looked between my two files and arbi­trar­i­ly chose to focus on the plan­et Par­a­digm first. Hon­est­ly, this is prob­a­bly because of my lin­ger­ing trep­i­da­tion at the dis­as­ter that was The Solar Sys­tem (expla­na­tion to come when I pub­lish the work). So, let’s build a planet.

Concept

Par­a­digm “the plan­et” is going to be the life-sus­tain­ing plan­et of this world. I want Par­a­digm to orbit a sun (suns) rather than anoth­er plan­et (as a satel­lite plan­et) because I like how it looks. Satel­lite plan­ets could rea­son­ably sus­tain life; a good friend offered Yavin 4 and Endor from Star Wars as good exam­ples of satellites.

Wants

I don’t want to devi­ate too far from Earth-like con­di­tions for Par­a­digm (at least on the planet’s sur­face). So the planet’s size (mass?) is going to be rough­ly the Earth’s. This note is impor­tant since I’d like grav­i­ty of Par­a­digm to be Earth-like. 

Hurdles

When I did this plan­et research, I found the hur­dle of “this is very advanced research for me”. Though out of my depth I still pressed on. And once I felt I had decid­ed every­thing I dou­ble-checked my work, only to learn that some­where I had made mis­takes (the math didn’t work out cor­rect­ly). So this time around, cor­rect­ing that error is my biggest hurdle.

The Process

Since I’m about to build a plan­et, I fig­ure some good space ambiance is required. Let’s strap on our astro­naut suits and get set to explore the unknown. Round 2 of build­ing a plan­et, here we go.

1. Get Scrap Paper. A lot of scrap paper. Relive (for bet­ter or worse) your mid­dle school math home­work days. I’m going to do my best to explain the math here.

Let’s start where I did, with Artifex­i­an and his world­build­ing videos. First up: ter­res­tri­al plan­ets. By ter­res­tri­al we mean rocky plan­ets: plan­ets that have a sil­i­cate or metal­lic core like the Earth’s. Com­pare this with gas plan­ets, ice plan­ets,  lava plan­ets, ocean plan­ets, or the the­o­ret­i­cal (undis­cov­ered) plan­et types.

2. Fol­low along with Artifex­i­an. I’ll put up time­stamps from his video when there’s infor­ma­tion to go over. I’ll still try to sum­ma­rize his infor­ma­tion here, but again I’m pulling a lot of these num­bers from his work.

First at (0:24) we’re giv­en Artifex­i­an’s plan­et­mak­er equa­tion and plan­et mass graph from Sea­ger et. al 2007’s paper (for a full expla­na­tion watch his first video on ter­res­tri­al plan­ets). Artifex­i­an gives us his equa­tion with some hab­it­able plan­et ranges (1:22):

${\mathrm{g}}_{\mathrm{e}} = \frac{{\mathrm{M}}_{\mathrm{e}}}{{{\mathrm{R}}_{\mathrm{e}}}^2} = {\mathrm{R}}_{\mathrm{e}}\left(\mathrm{\rho }\right)$

M_e = 0.1 <-> 3.5
R_e = 0.5 <-> 1.5
g_e = 0.4 <-> 1.6

g is the force of grav­i­ty, M the plan­et’s mass, R the radius, and ρ (rho) the plan­et’s den­si­ty. The sub­script _e is refer­ring to Earth: as an exam­ple M_e is the mass of Earth (rough­ly 5.972e²⁴ kg).
(I’m using sci­en­tif­ic nota­tion eⁿ to short­en “times the nth pow­er of ten” *10ⁿ)

Remem­ber here that these num­bers are all rel­a­tive to Earth, and are there­fore huge. 0.01 M_e = 5.9722e²² kg (50 sex­til­lion kilo­grams) and 0.01 R_e = 63.78 kilo­me­ters.
Mak­ing small changes to these num­bers (to get a plan­et that is Earth-like) will result in mas­sive differences.

Two notes here:

i. Yes I’m using met­ric. A lot of the physics we’re deal­ing with is in met­ric units, and I don’t want to con­vert between met­ric and impe­r­i­al mid-math.

ii. In terms of sig­nif­i­cant fig­ures and round­ing, my notes are going to be four or five dec­i­mal places. I’d rather have more dec­i­mals than too few.

Fin­ish­ing the video, we’ll be dis­mayed to see that this is all the infor­ma­tion we get. Ref­er­enc­ing the chart from Sea­ger et. al 2007’s paper (pg 21, fig. 4) is a lit­tle tough as well. So let’s work with what we have. 

Don’t for­get that the units we’re work­ing with are Earth-mass­es and Earth-radii. We can deal with these num­bers for now, but keep in the back of your mind that these are all mul­ti­plied by Earth’s rel­a­tive val­ue to find the actu­al size.

3. Pick some num­bers. Now it’s time to make deci­sions. We need the planet’s mass, radius, and grav­i­ty (in Earth units for now). Our con­straints for those three num­bers are as Artifex­i­an gave us: 

${\mathrm{g}}_{\mathrm{e}}=\frac{{\mathrm{M}}_{\mathrm{e}}}{{{\mathrm{R}}_{\mathrm{e}}}^2}= \mathrm{R}\left(\mathrm{\rho }\right)$

M_e = 0.1 <-> 3.5
R_e = 0.5 <-> 1.5
g_e = 0.4 <-> 1.6

If I wasn’t being com­pli­cat­ed enough with all of this, I want to start with one of my wants: I want Par­a­digm to be 25% larg­er than Earth. 25% larg­er here refers to the Earth’s vol­ume (V = 4/3 π r³), so I want a plan­et with 1.496 R_e.
(I solved the equa­tion for vol­ume in terms of r; feel free to just pick a number)

4. Check your work. Spot check 1.496 R_e: I want a plan­et with a vol­ume larg­er than the Earth, so the radius should be big­ger than the Earth’s radius (1 R_e). Good. This also just makes the radius cut­off above (R < 1.5), so I’m close but good.

The next part I’ll make easy for myself: the grav­i­ty will be the same as the Earth’s. I don’t want to mess with design­ing a world that has more or less grav­i­ta­tion­al force.
I can plug that into the “plan­et­mak­er equa­tion” (g = M/R² = R(ρ) ), using g = 1g_e (remem­ber this is “1 times Earth’s grav­i­ty”; it’s a unit in of itself). I get 2.239 M_e.

Spot check 2.239 M_e: a plan­et with a larg­er vol­ume and the same grav­i­ty needs to have a larg­er mass than Earth. If it was the same mass or less, the den­si­ty would be less and there­fore the grav­i­ty would be less (the same mass spread over a larg­er area). I’m still with­in the Earth-mass bounds (0.1 > M > 3.5), so it’s still look­ing good.

This all means for Par­a­digm ℗ I have:

M_p = 2.239 M_e
R_p = 1.496 R_e
g_p = g_e

If you’re curi­ous (like I am), in met­ric that’s:
(here I rec­om­mend using Wol­fram Alpha to do the math for you)

M_p = 1.337e²⁵ kg
R_p = 9.541e³ km   (9,541.488 km)
g_p = 9.807 m/sec²

Works-In-Progress

I’m afraid this post, like my last post, will be a bit long and bland. This kind of orga­ni­za­tion­al work isn’t very excit­ing. Hope­ful­ly in a post or two I’ll have the bor­ing work nailed down, and I can get to the fun and inter­est­ing stuff to write. Thanks for keep­ing up with me.