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How to Calculate the Density of the Solar System

Density is a well defined term, and the extent of the solar system is relatively well agreed upon, but putting those terms together creates an interesting problem. Density is total mass divided by total volume. The solar system is the sun and all the objects that are trapped by its gravitational influence --- although there are some differing opinions about the location of the boundary of the solar system. The density of a balloon (low), a cantaloupe (medium) or a cannonball (high) is easy to calculate. Because the solar system has so much empty space relative to the masses that occupy that space, density is hard to define. There are some reasonable approaches, though.

Instructions

Average Density
- 1
  
  The sun is by far the heaviest object in the solar system, but the rocky inner planets are all denser.
  
  Look up the density of each of the bodies in the solar system.
  
  Using the inner solar system as an example: the density of the sun is 1.4; Mercury is 5.4; Venus is 5.2; the Earth is 5.5; and Mars is 3.9; with all measured in grams per cubic centimeter.
- 2
  
  Mercury is the smallest planet, but it is rocky and dense.
  
  Add them all. For the densities of the inner solar system in the example:
  
  (1.4 + 5.4 + 5.2 + 5.5 + 3.9) = 21.4 g / cm^3.
- 3
  
  This radar map of Venus shows the rocky surface beneath the clouds.
  
  Divide by the number of objects. This is the average density of all the objects in your selected sample.
  
  So, for the inner solar system, the average density is: 21.4 / 5 = 4.3 g / cm^3.
Weighted Average Density
- 4
  Look up the mass and radius of each body in the solar system.
  
  Using the inner solar system as an example once again, the sun has a mass of 1,989,000 and a radius of 695,980. Mercury has a mass of .3 and a radius of 2440. Venus has a mass of 4.9 and a radius of 6052. Earth has a mass of 6.0 and a radius of 6371. Mars has a mass of 0.6 and a radius of 3390. The mass is in units of 10^24 kg and the radius is in kilometers.
- 5
  Add up the total mass. For the example, 1,989,000 + .3 + 4.9 + 6.0 + 0.6 = 1,989,011.8 x 10^24 kg, which equals 1,989,011.8 x 10^27 g.
- 6
  Calculate the volume of each body and add up the total volume. Volume = (4/3) * Pi * Radius^3.
  
  The volumes for the inner solar system example are sun = 1,412,145,200; Mercury = 61; Venus = 928; Earth = 1083; and Mars = 163 with all given in 10^24 cm^3. The total is 1,412,147,400 x 10^24 cm^3.
- 7
  
  All the planets together make up only a tiny fraction of the mass of the sun.
  
  Divide the mass by the volume. This would be the density of an object that was formed by shoving everything together in one giant blob: 1.989,011.8 * 10^27 g / 1,412,147,400 * 10^24 cm^3 = 1.4 g/cm^3. This is the same as the sun alone which makes sense because adding the planets into the sun is like tossing a couple pebbles into the sea.
Overall Mass Density
- 8
  Look up the masses of each of the objects in the solar system and add them together. Again take as an example the inner solar system with masses in units of 10^27 g: Sun=1,989,000; Mercury=0.3; Venus=4.9; Earth=6.0; and Mars=0.6 for a total of 1,989,011.8 * 10^27 g.
- 9
  
  The inner solar system extends out to Mars, the last of the rocky planets.
  
  Calculate the overall volume of the solar system. For the volume of the inner solar system, use a giant sphere all contained within the diameter of the orbit of Mars. The radius of Mars' orbit is 228 million kilometers. A sphere with that radius has a volume of 5 * 10^40 cubic centimeters.
- 10
  
  Most of the solar system is empty space --- even charts such as these exaggerate the size of the planets to make them visible.
  
  Take the total mass and divide by the overall volume of the solar system.
  
  This would be the density of an object created by mashing all the individual objects together, but then spreading the resulting glob out to fill all the space in the solar system.
  
  To finish the example:
  
  1,989,011.8 * 10^27 g / 5 x 10^40 cm^3 = 0.00000004 g / cm^3.
  
  This is an indication of just how empty the solar system is. This is more than 30,000 times less dense than air at the surface of the Earth.

Astronomy