# 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_{E }in 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.

Quantity | Symbol | Value |
---|---|---|

Earth mass | M_{E} |
5.97 × 10^{24 } kg |

Earth radius | R_{E} |
6.38 × 10^{6} m |

Jupiter mass | M_{J} |
1.90 × 10^{27} kg |

Jupiter radius | R_{J} |
7.15 × 10^{7} m |

Solar mass | M_{Sun} |
1.99 × 10^{30} kg |

Solar radius | R_{Sun} |
6.96 × 10^{8} m |

Distance between Earth and Sun | AU | 1.50 × 10^{11} m |

Now, instead of saying, for example, that the mass of Saturn is 5.68 × 10^{26} 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:

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

In this case:

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 _unit3.5.1 Activity 5 Sizes and masses of planets in terms of Earth and Jupiter

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.

Quantity | Symbol | Value |
---|---|---|

Earth mass | M_{E} |
5.97 × 10^{24} kg |

Earth radius | R_{E} |
6.38 × 10^{3} km |

Jupiter mass | M_{J} |
1.90 × 10^{27} kg |

Jupiter radius | R_{J} |
7.15 × 10^{4} km |

- Calculate the mass of Uranus in terms of Earth and Jupiter. Which comparison is more useful?
Uranus mass: 8.68 × 10

^{25}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 × 10

^{4}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 × 10

^{3}km

### Answer

0.034 R_{J} or 0.38 R_{E}; comparison with Earth is more useful.