Planet Creation: I'm so lost...

[Starwhip]

So I've been creating a realistic solar system for something I'm writing. I thought I had created a decent planet like Earth, but once again my calculations were off and it had a g-force higher than that of Jupiter.

As I emailed my friend:

" just rechecked some calculations, and Aurrera is so effed up it isn't even funny. With that density and the radius, it would have a g-force higher than that of Jupiter. And if I kept the original g-force and density, it would have to be approximately 610,000 Km in diameter, or 52.63 times larger than Earth. So I need to rethink this.

If we were to keep the radius and g-force, the density would be 2145.329 Kg/m^3, but that's about 2x less dense than the Earth.

SO COMPLICATED!!!

Well, let's see.

If D = M/V, then 6442.951 = M / V

And if M = 7.54113967 x 10^24, then 6442.951 * V = 7.5113967 x 10^24

Which gives V = 1.170448087 x 10^21

And if V = 4(pi)(radius^2)

(1.170448087 x 10^21) / 4(pi) = radius^2

9.192677776 x 10^20 = radius^2

3.031942904 x 10^10 = radius (m)

dividing by 1000 to give Km gives us 30319429.04 Km radius

I'm so friggin' lost it's painful..."

Any help? I still want that density and a reasonable g-force, but I think I'm going to have to have a different diameter.

Here's the equation for g force in m/s^2:

g = GM / r^2

where

g is the gravity of the planet at the surface in m/s^2

G is the gravitational constant: 6.67x10^-11

M is the mass of the planet/object in kg, and

r is the distance from the body (radius) in m

EDIT:

The density I wanted was 6442.951 Kg/m^3, since the planet would be rich in heavy materials like metals. The g-force should be between 0.9 g's (8.838 m/s^2) and 1.5 g's (14.73 m/s^2)

Edited April 7, 2014 by Starwhip

[K^2]

V = (4/3)pi r³ The formula you used is for surface area.

Always check your units. r² has units of m², and volume is m³. If you kept units through all of your operations, you'd see that you aren't getting radius in meters.

[RRG]

You used the area equation instead of the volume equation.

Area=4(pi)R²

Volume=4/3(pi)R³

So:

a=MG/r²

a=vdG/r²

v=ar²/dG

v=4/3*pi*r³=ar²/dG

r=((3/4)*a)/(dG*pi)

a=Acceleration due to gravity

M=Mass

r=Radius

d=Density

G=Gravitational constant

[Ralathon]

Your friend messed up the formula for the volume of a sphere, he's calculating surface area instead. The volume of a sphere as a function of its radius is:

V = (4*pi*r³)/3

This gives it a mass of M = rho*V. (rho = 6442.951 [kg*m^-3])

Now, we just insert this into Newtons law of gravitation and simplify to:

g = G*rho*(4*pi*r³)/(3*r²)

= G*rho*4*pi*r/3

~= 1.80142e-6 * r

Where r [m] is the radius of the planet and g [m*s^-2] is your surface acceleration

So, say you want a surface acceleration of 10 [m*s^-2] you'll end up with a radius of 5.551e6 meters, or about 5500 kilometers. Which is about a 1000 kilometers smaller than earth.

[Starwhip]

No, shoot, that was me.

One mo, I'll recheck my calculations.

Ok, so:

11.76 = ((6.67 x 10^-11)(7.5113967x10^24)) / r^2

11.76r^2 = 5.010101599 x 10^14

r^2 = 4.260290475 x 10^13

r = 6527090.068 (meters)

r = 6527.09 (Km)

which means diameter = 13054.18 Km

That's pretty damn close to the 13080 Km that I had originally! Huzzah!

Now density.

D = M/V

V = (4/3)pi radius(meters)^3 = 1.164789408 x 10^21 m^3

M / V = 6448.716524 Kg/m^3

It works again!

Edited April 7, 2014 by Starwhip