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Lesson 5 of 8Space Exploration
Unit 5 · Lesson 5 of 8

Distances and Scale in Space

Space distances are so huge that kilometres become awkward. Use scientific notation to handle the big numbers, the astronomical unit (AU) for distances in the solar system, and the light-year for distances to stars and galaxies, then build a scale model to picture how empty space really is.

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Key things to remember

  1. 1Scientific notation: with . Multiply: add exponents. Divide: subtract exponents.
  2. 21 AU = average Earth–Sun distance km (about 150 million km).
  3. 31 light-year = distance light travels in one year km (about 63 000 AU).
  4. 4Speed of light km/s; time ; sunlight takes about 8.3 min per AU.
  5. 5Scale model: model size real size using the same factor for every size and distance.

The nearest star to the Sun, Proxima Centauri, is about 40 000 000 000 000 km away: forty trillion kilometres. Numbers like that are hard to write, hard to read and easy to get wrong by a zero or two. Astronomers solve this in two ways. First, they write big numbers in scientific notation. Second, they use bigger units of distance: the astronomical unit inside the solar system and the light-year for stars and galaxies. Just as you would measure a pencil in centimetres but a road trip in kilometres, you pick the unit that keeps the numbers small.

Scientific notation

A number in scientific notation is written as a number from 1 up to (but not including) 10, multiplied by a power of 10. The power tells you how many places the decimal point moves. To write 149 600 000 in scientific notation, put the decimal point after the first digit to get 1.496, then count how many places the decimal point moved to the left: 8. So 149 600 000 = . A positive exponent means a big number; each step up in the exponent means ten times bigger. To multiply numbers in scientific notation, multiply the front numbers and add the exponents. To divide, divide the front numbers and subtract the exponents. If the front number ends up 10 or more (or less than 1), shift the decimal and adjust the exponent.

Left: writing a big number in scientific notation. Right: multiplying, then fixing a front number that is bigger than 10.

The astronomical unit (AU)

One astronomical unit (AU) is the average distance from Earth to the Sun: km, or about 150 million km. It is the natural ruler for the solar system. Mars is about 1.5 AU from the Sun, Jupiter about 5.2 AU and Neptune about 30 AU. Distances in AU are easy to compare at a glance: Neptune is about 30 times as far from the Sun as Earth is.

Distance from the SunSun0.5125102050AU, logarithmic scale (1 to 10 takes the same space as 10 to 100)Asteroid belt2.2–3.2 AUKuiper belt30–50 AUMercury0.39 AUVenus0.72 AUEarth1.00 AUMars1.52 AUJupiter5.20 AUSaturn9.54 AUUranus19.2 AUNeptune30.1 AUTerrestrial (rocky)Gas giantsIce giantsRelative sizes (to scale with each other, not with the distances above)MercuryVenusEarthMarsJupiterSaturnUranusNeptune
Fig 1The planets in order with their average distances from the Sun in astronomical units on a logarithmic scale, where each step of ten times takes the same space, from Mercury at 0.39 AU to Neptune at 30.1 AU, with the asteroid belt and Kuiper belt marked; planet sizes are shown in a separate row because sizes and distances cannot be drawn to the same scale.

The light-year

Even the AU is too small for the stars: Proxima Centauri is about 268 000 AU away. For stars and galaxies we use the light-year (ly): the distance light travels in one year. Light is the fastest thing in the universe, moving at about km every second. A year has about seconds. Multiplying gives the length of one light-year. Even though it has the word "year" in it, a light-year measures distance, not time.

One light-year is about 9.46 trillion kilometres, or about 63 000 AU.
ObjectDistance from EarthIn kilometresTime for its light to reach us
The Moon384 400 kmabout 1.3 seconds
The Sun1 AUabout 8.3 minutes
Neptune (from the Sun)30.1 AUabout 4.2 hours
Proxima Centauri (nearest star after the Sun)4.24 ly4.24 years
Centre of the Milky Wayabout 26 000 lyabout 26 000 years
Andromeda Galaxyabout 2.5 million lyabout 2.5 million years
From the Moon to the nearest large galaxy. Each step outward needs a bigger unit to keep the numbers manageable.
Looking far away means looking back in time
Light takes time to travel, so you always see objects as they were when the light left them. You see the Moon as it was 1.3 seconds ago and Proxima Centauri as it was about 4 years ago. The faint smudge of the Andromeda Galaxy, visible without a telescope from a dark site, is light that left about 2.5 million years ago. Telescopes that see very distant galaxies are seeing the young universe.

Scale models

A scale model shrinks every real size and distance by the same factor. Suppose the Sun (real diameter km) is a 30 cm beach ball. Then every real length is multiplied by . On that scale Earth is a peppercorn under 3 mm wide, about 32 m from the ball: roughly a third of a football field. Mars is about 49 m away, Jupiter (a 3 cm marble) about 168 m away, and Neptune almost 1 km away. And Proxima Centauri? About 8700 km away, farther than the distance from Toronto to London, England. Space is mostly empty.

Scientific notation on a calculator
Enter with the EE or EXP key: type 1.496, press EE, then type 8. Typing "1.496 × 10 ^ 8" also works, but in a division put it in brackets, or the calculator may divide by 1.496 and then multiply by 10 to the power 8. Always check that your answer has a sensible size.

Worked examples

Reveal one step at a time and try to beat it to the next line.

Example 1
Jupiter is 5.20 AU from the Sun. Convert this distance to kilometres in scientific notation.
Example 2
Proxima Centauri is 4.24 light-years away. How far is that in kilometres?
Example 3
In a scale model the Sun is a ball 30 cm wide. The real Sun is km wide. How far from the ball should you place Mars, which is 1.52 AU from the Sun? Give the answer in metres.
Example 4
A star is km away. How many light-years is that?

Where marks get lost

  • Treating a light-year as a time. It is a distance: how far light travels in a year (about 9.46 trillion km).
  • Leaving the front number outside 1 to 10, such as . Rewrite it as , adding 1 to the exponent.
  • Adding exponents when dividing. When you divide powers of 10, subtract the exponents; add them only when you multiply.
  • Mixing units in a scale model, such as dividing a distance in km by a size in cm without keeping track. Make sure the units cancel or convert first.
  • Thinking the planets are close together because diagrams show them that way. Most pictures shrink distances far more than sizes; on a true scale the planets are tiny dots far apart.

Flashcards 7

Scientific notation
Astronomical unit (AU)
Light-year (ly)
Speed of light
+ 3 more

Practice

1
A light-year is a unit of…
2
Which is 384 400 km (the distance to the Moon) written in scientific notation?
3
An asteroid is 3.5 AU from the Sun. How far is that in millions of kilometres? (1 AU = 149.6 million km.) Round to the nearest million.
million km
4
Sirius, the brightest star in the night sky, is about 8.6 light-years away. How far is that in trillions of kilometres? (1 ly = 9.46 trillion km.) Round to the nearest trillion.
trillion km
5
In a scale model the Sun is a 20 cm ball. The real Sun is km wide and Earth is km from the Sun. How far from the ball should Earth be placed, in metres? Give your answer to one decimal place.
m
6
A rover on Mars is 2.0 AU from Earth. Radio signals travel at the speed of light, about 8.3 minutes per AU. How many minutes does a command from Earth take to reach the rover? Give your answer to one decimal place.
min
7
Put these in order from closest to Earth to farthest from Earth.
  1. 1Centre of the Milky Way
  2. 2Proxima Centauri
  3. 3Andromeda Galaxy
  4. 4The Moon
  5. 5The Sun
  6. 6Neptune
8
Match each distance to the most sensible unit for measuring it.
Length of a soccer field
Drive from Toronto to Ottawa
Distance from the Sun to Jupiter
Distance from the Sun to Sirius
9
How many astronomical units are in one light-year? Divide km by km and round to the nearest thousand.
AU

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