# How to calculate the density contribution of each of Earths layers

I'm trying to calculate the density of planets a bit more precise, than just treating it like an object of uniform density.

In order to do so, I went with Earth, as my convenient sample planet. Earth consists of several distinct 'layers', each with their own (average) density, that combine into an average density of for the entire planet of 5,507 kg/m³.

However, when working with the data as listed in the table below, I get a total average density of 6584.39 kg/m³ (= the sum of the last column), which is obviously incorrect.

Layer density (kg/m³) thickness (km) volume (km³) volume (%) Mass (kg) ? kg/m³
Ocean (60%) 1,000 3.688 353,861 0.07% 353,861,487 0.69
Oceanic crust (60%) 2,500 10 958,462 0.19% 2,396,155,549 4.69
Continental crust (40%) 2,500 35 2,240,810 0.44% 5,602,025,188 10.98
Upper mantle 3,900 390 60,339,690 11.82% 235,324,792,761 461.04
Transition zone 3,300 250 36,668,669 7.18% 121,006,609,194 237.07
Lower mantle 5,000 2,234 257,937,223 50.53% 1,289,686,113,497 2526.71
Outer core 11,000 2,261 133,340,163 26.12% 1,466,741,789,259 2873.60
Inner core 12,900 1,216 18,581,339 3.64% 239,699,277,062 469.61

So, the question is

what am I doing wrong?

Answer: My volume calculations were wrong. Fixed table for future reference:

Density (kg/m³) Thickness (km) Volume (km³) Volume (%) Mass (kg)
Ocean (60%) 0.69 4.383 1,336,885,460 0.12% 922,450,967
Oceanic crust (60%) 2,500 10.349 3,149,560,997 0.29% 7,873,902,493,345
Continental crust (40%) 2,500 35.362 7,202,736,622 0.66% 18,006,841,555,992
Upper mantle 3,900 390 185,606,710,958 17.13% 723,866,172,736,338
Transition zone 3,300 250 106,905,562,191 9.87% 352,788,355,229,284
Lower mantle 5,000 2,231 602,928,441,022 55.66% 3,014,642,205,109,740
Outer core 11,000 2,261 168,545,384,581 15.56% 1,853,999,230,387,040
Inner core 12,900 1,216 7,531,636,199 0.70% 97,158,106,969,204
5,602.19 6,370.98 1,083,206,918,030 100.00% 6,068,335,736,931,910
• Earth Science tends to get confused when you come in from a world building perspective. Their frame of mind when answering questions is firmly rooted in the real-world Earth. Commented May 2, 2023 at 13:17
• If the mods feel this question is more applicable to Earth Science.SE, I guess it should be moved. Commented May 2, 2023 at 13:30
• I’m not a mod, so the ball is in their court. Commented May 2, 2023 at 13:32
• Where did you get your table from? Ocean density 0.69 kg/m3 ??? I got the same result as you, based on your table, but GIGO
– azk
Commented May 2, 2023 at 15:15
• You should role your table back to its original state, or else answers to your original question no longer make any since. Commented May 2, 2023 at 17:39

You don't just add up the densities.

Let's say you add 1 cubic meter of water, which has a mass of 1000 kg, and 1 cubic meter of olive oil, which has a mass of 916 kg.

What you have done by adding up the densities is:

$$\rho_{mix}=\rho_{water} + \rho_{oil}= M_w/V_w + M_o/V_o=(M_wV_o+M_oV_w)/V_oV_w$$

What you should have done is, considering that you have 2 cubic meters of liquid with a mass of 1916 kg:

$$\rho_{mix}=(M_w + M_o)/(V_w + V_o)$$

Which as you can see is a different result.

• I've added a Mass column (=density * volume). If I sum the mass column, and divide it by the sum of the volumes... I get the exact same answer as I got before :( I feel like an idiot for not being able to follow your clear instructions. Commented May 2, 2023 at 13:26

First of all, your source data table is all wrong. Begin by converting your thickness to meters. Otherwise, all of the rest of your data will be messed up by conversion errors.

Secondly, your volumes make no since. The volume of a sphere is 4/3πr^3. For simplicity, you can represent 4/3π as 4.1887867 in your formulas. So, if you are using Excel, your inner core's volume should be =(C9)^3*4.1887867, then your outer core would be =(C8+C9)^3*4.1887867-E9 and your lower mantle =(C7+C8+C9)^3*4.1887867-E8-E9 etc...

Your top 3 layers are a bit strange because they don't all stack normally. Both crusts form spheres that add to the radius of the Upper Mantle and then the Ocean is added to the radius of the Oceanic Crust

Once your data table is cleaned up, it should look like this:

If you look up any of these values volumes, they may not be exactly the same between sources, but you should be within the right order of magnitude assuming your sources for how thick things are were reliable. The goal here is to have a process to estimate an alien world, not account for every mountain and valley here on Earth.

## Once, your Data is fixed, it's very easy

Finally, to find the average density you just need to divide the total Mass by the total Volume which will yield an average density of 5564kg/m^3

• If they are using Excel, or Google Sheets, or LibreOffice Calc, then π is PI(). Commented May 2, 2023 at 19:47