Calculating Mass/Density for a 4x Larger Earth

[surfaceGravity]

I'm trying to figure out what the mass or the average density of the Earth would be if it were made of the same material, but more of it, to the point that its radius were 4 times larger. My ultimate goal is to use this information to predict the surface gravity (relative to that of our current Earth) that one would encounter if the Earth were 4 times as wide, according to the equation Surface Gravity = (Mass (4r Earth) / Mass (1r Earth)) / (4^2). However I've run into a complication.

I understand that as things (planets) increase in size, they compress under their own weight, meaning that the Earth would have a lower average density if it were smaller and less massive, and would have a higher average density if it were larger and more massive. As such, I would expect that it's mass would not increase linearly in relation to a change in radius.

I understand that Density = Mass / Volume, and that M = D * V. However, according to this equation, one always needs either mass or density in order to calculate the other. While figuring out the Volume of "4r Earth" is easy, I have neither density nor mass, and am not sure how to determine either, independently.

What I need is a way to calculate either mass or average density based on the radius alone, by predicting the extent to which the planet would compress under its own weight as the radius increased. Is there some kind of equation that can describe the compression of a planet as it increases in size, so that one can determine the true mass or average density that would result from an increase in its radius? We can assume that the planet is a perfect sphere.

I know that there are different layers within the Earth, with different elemental compositions. I'm not sure to what extent this would play a factor, but I would expect that different substances compress to different extents, as pressure increases. If need be, perhaps we can assume a uniform composition throughout.

Any help or advice is welcome. I honestly don't have a clue what to do.
Feel free to let me know if we need more info and I'll do what I can to hunt it down.
Thanks!

 

[madphysics]

How can you find the radius of the Earth if it is not a real sphere? It is slightly flattened at the poles and rounder at the equater due to centrifical force.

 

[russ_watters]

That isn't a significant factor for this problem, madphysics.

surfaceGravity, though what you are saying is true, I wouldn't think it would be a significant issue for your purposes. For example, the moon is roughly 3/5 as dense as the earth, despite being 1/4 the radius. But that is probably mostly due to the difference in composition: the Earth's core is it's densest part and the moon is essentially a broken-off piece of the mantle.

Because solid planets are solid, compressibility is relatively low. This is, however, a significant issue for the gas giants.

 

[surfaceGravity]

To give this discussion some more depth I went roaming and roving through the net, and I eventually came upon a PDF file which seams to summarize the observed and predicted values for planets with varying radii and compositions.

I haven't had the time to read it yet, but on page 22 there is a tidy little graph depicting radius mapped against mass. If I understand the graph correctly, it seems that for a planet like the Earth, 4x the radius is roughly the maximum size possible. From that point onwards it appears to me, on reading the graph, that the sheer mass of the planet prohibits it from gaining significantly in size (due to compression under pressure), even as mass increases. But I could be reading it incorrectly, or perhaps by reading the rest of the paper it would be put in better perspective.

There is also a percentage error in mass estimate scale which I don't understand. Would some one be willing to look at the document linked below, to comment on what the graph is saying and if possible, explain to me how the percentage error scale should be interpreted?

The PDF can be found here: http://www.citebase.org/fulltext?format=application%2Fpdf&identifier=oai%3AarXiv.org%3Aastro-ph%2F0612671
If the link should ever become broken, the title of the paper is: Planetary Radii across Five Orders of Magnitude in Mass and Stellar Insolation: Application to Transits
by: J. J. Fortney, M. S. Marley, and J. W. Barnes

Thanks for the replies so far! :)

 

[russ_watters]

I guess the difference is larger than I thought - the pressures are very high. Page 19 shows density vs pressure curves for various materials. You could integrate that with depth/radius to determine the average density for any planet size in that range.

Ie, for water the relationship is 10m per bar initially, so to get you onto that graph, you need to go down 1km, to a pressure of 100bar. It'll take some work, but I think you can curve fit that graph and then numerically integrate this in excel using depth as radius and a thin shell approximation for the total mass of each slice/shell at a certain depth.

 

[surfaceGravity]

Woah, I think that's out of my league. :) Maybe I should just trust the graph. The graph is good. The graph knows all. lol

But if I knew how to interpret the "percentage error" part of the graph, (in the upper left quadrant), then perhaps I'd feel better about letting the graph speak for itself. Any advice on how to apply the "percentage error" to the graph? ie: what's it saying? :)

And is the graph making assumptions about the internal temperatures of the planets? I would think that a hotter interior would decrease density and increase radius, to some extent.