### road blocks

Well, I said I'd post about the cousins today. Problem is that I didn't get around to working out a few little parts about them, like what they look like or what their home system is like. I got stuck on some work that I was doing trying to redo some work I've done before and lost about the Sarthaki, mostly working out the size of the core of the planet as well as the size of the mantle. It is a fairly simple project, but my algebra skills are a little rusty. Too many years of solving systems of equations by just plugging them all into my TI89 and telling the graphing function to find the intercepts. Lost the calculator a year ago, still haven't bothered to relearn the stuff I've forgotten. Here's the situation.

The Sarthaki homeworld has a radius about 1.73 time that of the Earth, and a surface gravity 1.2 times that of Earth. You can get the gravity of a planet with the formula Gp=Rp*(Pp/Pe) Where Rp is the radius of the planet, Pp is the density of the planet (I use grams/cm^3) and Pe is the density of the Earth. this time I know Gp, Rp, and Pe (1.2, 1.73, and 5.5g/cm^3). Solving for Pp, we get a density of about 3.8 g/cm^3. Now a rocky planet like the Earth has two major components: A metal (iron) core with a density of about 10g/cm^3 and a rocky mantle/crust complex with a density of about 2.5 g/cm^3. On Earth, the core is about 55 percent of the radius of the planet (and if you knocked off all of the lighter surface material, the gravity at the surface of the core would be about 99 percent of the gravity at the surface of the Earth, but that isn't too important.)

This is where it gets a little fuzzy.

To get the sizes of the two layers, we can use this equation:

3.8= (10*m+2.5*n)/(m+n)

m is the volume of the core and n is the volume of the mantle.

in our case, m+n is approximately 21.7 (it is exactly 1.73^3*pi*4/3, and the units are Earth radi cubed)

At this point, there is a choice. Do I solve the equation as it stands, or do I abstract it out a little further first. If I were using a calculator, I'd definately go the next step because it puts everything into a single variable.

we have:

m=4/3*pi*b^3

and

n= (4/3*pi*1.73^3)-m (n is a spherical shell with a m sized hole.)

so the equation becomes this monstrocity

3.8= (10*(4/3*pi*b^3)+2.5*((4/3*pi*1.73^3)-(4/3*pi*b^3)))/21.7

To solve, I would plug in the equations

y=(10*(4/3*pi*x^3)+2.5*((4/3*pi*1.73^3)-(4/3*pi*x^3)))/21.7

and

y=3.8

into the grapher and aks for the intercepts.

which would give the radius of the core.

Not that any of this is horribly important stuff since I already have the average density and the surface gravity, and the radius, but I do like to know these things.

But since I was working on that (and playing Sim City 2000) I didn't get anything done with the Malar.

I do have a fair amount of social and some historical information, but it needs at least their gross anatomy to round it out, and I am not even sure what sort of world they live on.

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