5 Simplifying the numbers

You’ve seen in the last few sections that studying planets involves dealing with some very big numbers. To simplify calculations, astronomers have introduced their own units, and you’ve already seen most of the important units for this course.

The sizes of planetary systems are measured in terms of astronomical units, written AU.
Giant planets are measured in terms of the mass and radius of Jupiter, written M_J and R_J.
Terrestrial planets are measured in terms of the mass and radius of the Earth, written M_E and R_E. Sometimes the alternative forms M_⊕ and R_⊕ are used, but you will use M_E and R_Ein this course.
Stars are measured in terms of the mass and radius of the Sun: the solar mass and solar radius, written M_Sun and R_Sun. Sometimes the alternative forms M_⊙ and R_⊙ are used, but you will use M_Sun and R_Sun in this course.

Table 2 Units used in exoplanet science

Quantity	Symbol	Value
Earth mass	M_E	5.97 × 10²⁴ kg
Earth radius	R_E	6.38 × 10⁶ m
Jupiter mass	M_J	1.90 × 10²⁷ kg
Jupiter radius	R_J	7.15 × 10⁷ m
Solar mass	M_Sun	1.99 × 10³⁰ kg
Solar radius	R_Sun	6.96 × 10⁸ m
Distance between Earth and Sun	AU	1.50 × 10¹¹ m

Now, instead of saying, for example, that the mass of Saturn is 5.68 × 10²⁶ kg, you can express the mass of Saturn in terms of the mass of Jupiter, M_J. To do this you need to work out how many Jupiter masses there are in Saturn. Mathematically, divide the mass of Saturn by the mass of Jupiter and the answer is the number of Jupiter masses in Saturn.

So:

\begin{align} \text{mass of Saturn} &= \frac{5.68 \times 10^{26}~\mathrm{kg}}{1.90 \times 10^{27}~\mathrm{kg}}~\mathrm{M}_{\mathrm{J}}\\ & = 0.3~\mathrm{M}_{\mathrm{J}}\end{align}

Or instead you could express Saturn’s mass in terms of the mass of the Earth.

In this case:

\begin{align} \text{mass of Saturn} &= \frac{5.68 \times 10^{26}~\mathrm{kg}}{5.97 \times 10^{24}~\mathrm{kg}}~\mathrm{M}_{\mathrm{E}}\\ & = 95~\mathrm{M}_{\mathrm{E}}\end{align}

So, you can say that Saturn’s mass is 0.3 M_J, or 95 M_E. Both of these alternatives give you an immediate feeling for where Saturn fits compared with other planets.

Activity 5 Sizes and masses of planets in terms of Earth and Jupiter

Allow about 10 minutes

You are given the masses and radii for some of the Solar System planets. You need to calculate the masses in terms of M_J and M_E, and the radii in terms of R_J and R_E. For each planet, which is the most sensible comparison to use?

Useful values are provided in Table 3. Note that all radii are given in km here.

Table 3 Units used in exoplanet science

Quantity	Symbol	Value
Earth mass	M_E	5.97 × 10²⁴ kg
Earth radius	R_E	6.38 × 10³ km
Jupiter mass	M_J	1.90 × 10²⁷ kg
Jupiter radius	R_J	7.15 × 10⁴ km

Calculate the mass of Uranus in terms of Earth and Jupiter. Which comparison is more useful?
Uranus mass: 8.68 × 10²⁵ kg

Answer

0.046 M_J or 14.5 M_E; either comparison is useful.

Calculate the radius of Neptune in terms of Earth and Jupiter. Which comparison is more useful?
Neptune radius: 2.48 × 10⁴ km

Answer

0.35 R_J or 3.9 R_E; either comparison is useful.

Calculate the radius of Mercury in terms of Earth and Jupiter. Which comparison is more useful?
Mercury radius: 2.44 × 10³ km

Answer

0.034 R_J or 0.38 R_E; comparison with Earth is more useful.

◄Contents►Next: 6 Gallery of planet portraits

OpenLearn - An introduction to exoplanets
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