View Full Version : A Question About Surface Gravity.

BigDon

2009-Feb-04, 07:35 AM

I'm not sure where but I am having an issue with why (oooh I have to articulate this right) surface gravity doesn't seem to scale up linearly. Two Earth volumes, at the roughly the same Earth density, doesn't give you a surface gravity of 2, but somewhere lower.

Am I reading this right or am I having post seisure moment?

Well surface gravity = GM/r²

Earth's radius is 6378 km. "Two earth volumes" at the same density (5500 kg/m³) gives you 2 earth masses, but it corresponds to a radius of 8035 km. So in this case you're not doubling the radius to get that mass, and the surface gravity is actually 1.26g.

If you wanted to double the gravity with the same density, you'd need to double the radius to 12756km, which would give you a mass of about 8 earth masses.

BigDon

2009-Feb-04, 08:35 AM

Thank you EDG,

You've hit it squarely on the head!

That's exactly what I don't get.

Jens

2009-Feb-04, 08:45 AM

That same things puzzled me a long time ago, when I found out that the moon is something like 1/16 the mass of the earth, but the gravity is like 1/3 (and I think the densities are similar). I wondered why, but it makes sense, because when you're on the moon, you're much closer to the center of the mass. I think that if you built a ladder on the moon that was high enough to bring you out to the radius of the earth in distance, the gravity would be much lower.

Thank you EDG,

You've hit it squarely on the head!

That's exactly what I don't get.

I found a nice explanation on wikipedia that may be a bit clearer. They point out there that density = (mass/volume) - so mass = density * volume = rho*(4/3)*pi*r³. So put that into the gravity equation and you get:

gravity = [G.rho.(4/3).pi.r³] / r²

you can cancel r² out from top and bottom and you'll get:

gravity = G.rho.(4/3).pi.r

That means that gravity is directly proportional to radius if the density is the same. So if you keep the density the same, you have to double the radius to get double the gravity. What you did was double the volume, which is not the same as doubling the radius - volume is proportional to the cube of the radius. If you check, you'll see that 8035 km for your "two earth volume" world is equal to CUBEROOT(2)*6378 km, which proves the point. And also, our 'two gravity" world that was double the radius and the same density has eight times the mass, and 2 * 2 * 2 = 8. So that all works out.

Does that clarify it for you?

( see http://en.wikipedia.org/wiki/Surface_gravity for the wiki article I mentioned)

Spaceman Spiff

2009-Feb-04, 04:04 PM

I found a nice explanation on wikipedia that may be a bit clearer. They point out there that density = (mass/volume) - so mass = density * volume = rho*(4/3)*pi*r³. So put that into the gravity equation and you get:

gravity = [G.rho.(4/3).pi.r³] / r²

you can cancel r² out from top and bottom and you'll get:

gravity = G.rho.(4/3).pi.r

What's also cool is that the cooling time scale of a planet scales in exactly the same way (all else other than density and radius being equal), because the thermal energy content scales as M and the surface area which ultimately dictates the flow rate of radiative energy into space scales as R^2.

BigDon

2009-Feb-04, 08:31 PM

I found a nice explanation on wikipedia that may be a bit clearer. They point out there that density = (mass/volume) - so mass = density * volume = rho*(4/3)*pi*r³. So put that into the gravity equation and you get:

gravity = [G.rho.(4/3).pi.r³] / r²

you can cancel r² out from top and bottom and you'll get:

gravity = G.rho.(4/3).pi.r

That means that gravity is directly proportional to radius if the density is the same. So if you keep the density the same, you have to double the radius to get double the gravity. What you did was double the volume, which is not the same as doubling the radius - volume is proportional to the cube of the radius. If you check, you'll see that 8035 km for your "two earth volume" world is equal to CUBEROOT(2)*6378 km, which proves the point. And also, our 'two gravity" world that was double the radius and the same density has eight times the mass, and 2 * 2 * 2 = 8. So that all works out.

Does that clarify it for you?

( see http://en.wikipedia.org/wiki/Surface_gravity for the wiki article I mentioned)

Thank you Edg. You get today's "Educator of Fellow Man" award.

Now pardon me while I have a "Black Monolith" moment. :D

Nick Theodorakis

2009-Feb-04, 08:49 PM

...Now pardon me while I have a "Black Monolith" moment. :D

Put down that bone before you hurt somebody!

Nick

BigDon

2009-Feb-04, 09:29 PM

But Nick!

If I don't, we will still besitting around our monitors a million years from now!

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