An introduction to exoplanets
An introduction to exoplanets

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An introduction to exoplanets

4.6  Average densities

If a planet is composed of different layers each having different densities, then the density of the planet as a whole will be averaged over the various layers. To give the average density of a planet, the total mass is divided by the total volume:

density equation sequence equals total mass divided by total volume equals cap m divided by cap v

In this way you can measure the average densities of some planets orbiting around other stars even though you can’t see them, as you will learn in future weeks. If a planet has the density of rock, it is probably composed of rock. If it has the density of hydrogen gas, that is probably what it is composed of. Iron is the densest material you might expect to find in a planet, while a mixture of hydrogen gas with a small amount of helium is the least dense material you might expect to find.

However, you must be careful about conclusions such as those above. The average density of a planet might not tell you what the planet is made of. It is possible to imagine a planet with an iron core surrounded by hydrogen/helium gas, which averages out to exactly the density of rock. In this case, the conclusion that the planet is composed of rock would be completely wrong. From the average density, it is impossible to be sure exactly how much of each layer is present.

As you go down to deeper layers of the planet Jupiter, you can see how much the density increases (Figure 12). Density is often expressed in units of grams per cubic centimetre (g/cm3), so the numbers in Figure 12 say how many grams of material are contained in each cubic centimetre. Near the core the density is expected to be around 25 g/cm3, which means that each centimetre cube contains material that has a mass of 25 grams. As a comparison, the density of liquid water on Earth is 1 g/cm3.

Described image
Figure 12  Density of Jupiter’s layers

Activity 3  The density of a layered planet

Timing: Allow about 15 minutes

In this activity you will calculate the density of a layered planet, in this case, Jupiter, which is shown in Figure 12.

  1. On a piece of paper, write down a sum giving the total mass M of Jupiter in terms of the masses of its four layers, M1, M2, M3 and M4.

Answer

cap m equals sum with, 4 , summands cap m sub one plus cap m sub two plus cap m sub three plus cap m sub four
  1. Write down an expression for the volume of Jupiter in terms of its radius, RJ.

Answer

volume equation left hand side equals right hand side 4.19 times normal cap r sub normal cap j cubed

Look back at Section 4.2 if you are unsure where this comes from.

  1. Referring to your answers to Part 1 and Part 2, write down an expression for the density of Jupiter in terms of M and RJ.

Answer

density equation left hand side equals right hand side cap m divided by 4.19 times normal cap r sub normal cap j cubed
  1. Use your answer to Part 3 and the data listed below to calculate the average density of Jupiter.

    Total mass of Jupiter: 1.898 × 1030 g

    Average radius of Jupiter: 6.991 × 10 9 cm

Answer

We use the expression for density found in Part 3 and substitute the numbers for the mass and average radius into it. This gives

multiline equation row 1 density equation left hand side equals right hand side cap m divided by 4.19 times normal cap r sub normal cap j cubed row 2 equation left hand side equals right hand side 1.898 multiplication 10 super 30 divided by 4.19 postfix multiplication left parenthesis 6.991 multiplication 10 super nine times right parenthesis cubed row 3 equals 1.3 g solidus cm super three

Jupiter’s average density is 1.3 g/cm3, which means that each centimetre cube of material has a mass of 1.3 grams on average – a little more than the density of water on Earth.

Note that the answer to a calculation like this should be rounded to convey how precisely the measurements used in the calculation are known. There are rules for this, but in this course it is generally suitable to round to two or three digits.

Help

Maths help

The following OpenLearn resources may help with the maths in this section. Open the link in a new tab or window by holding down Ctrl (or Cmd on a Mac) when you click on the link. Remember to return here when you have finished.

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