Units for astronomical distances

​​In a nutshell

Astronomical distances are so large that measuring them in metres is unhelpful. Instead, specialist units are used which are the astronomical  unit, light-year and parsec. Stellar parallax is used to calculate the distance from the Earth to a relatively near star. 

​Equations

Constants

The astronomical unit (AU)

The astronomical unit​ is the distance between the Earth and the sun, which is $$1.50 × 10^{11} m$$. It is mainly used to measure distances between planets and the Sun.​

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The light-year (ly)

The light-year​ is the distance travelled by light in a vacuum in one year, which is $$9.46 × 10^{15} \,m$$. This is calculated by:

 $$distance = speed \times time \newline \\[0.1in] distance = 3 × 10^8ms^{-1} × 365 × 24 × 60 × 60 \newline \\[0.1in]1 \space light-year = 9.46 × 10^{15} \, m$$

Example 

A meteor is approximately $$5.35 \ light-years$$​ from Earth. Calculate the distance between the Earth and the meteor in astronomical units.

State variables:

$$1\ light-year = 9.5 × 10^{15} \, m$$

$$1 \ AU = 1.496 × 10^{11}\, m$$​​

Distance to the meteorite = $$5.35 \,ly$$

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Convert into metres​:

​$$5.35 \, ly = 5.35 × (9.5 × 10^{15}) \newline 5.35 \, ly = 4.82 × 10^{16} \, m$$

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Convert from metres to AU​:

$$\dfrac{4.82 \times 10^{16}}{1.496 \times 10^{11}} = 3.2 × 10^{5} AU$$ (to 2sf)

The meteor is $$\underline{ 3.2 × 10^{5} AU }$$ from the Earth.

The parsec (pc)

The parsec is the distance that a radius of AU​ subtends an angle of $$1 \,arcsecond$$. Angles in astronomy are measured in $$arcminutes$$​ and $$arcseconds$$​ instead of degrees, there are $$60 \, arcminutes$$ in $$1\degree$$ and $$60 \, arcseconds$$ in an arcminute:

 $$1 \space arcminute = (\dfrac{1}{60})^\degree \newline \\[0.05in]1 \space arcsecond = (\dfrac{1}{3600})^\degree$$

Using the triangle above, the value of a parsec​ can be calculated as below:

​$$tan(arcsecond) = \dfrac{1 AU}{1 pc} \newline \\[0.1in]1 pc =\dfrac{1.50 \times 10^{11}}{tan(\dfrac{1}{3600})} \newline \\[0.1in]1 pc= 3.1 \times 10^{16}\,m$$

Example 

A meteor is approximately $$5.35 \ light-years$$​ from Earth. Calculate the distance between the Earth and the meteor in parsecs.

State variables:

$$1 \ parsec ≈ 3.1 × 10^{16} \,m$$​​

$$5.35 \, ly = 5.35 × (9.5 × 10^{15}) \newline 5.35 \, ly = 4.82 × 10^{16} \, m$$​

Convert from metres into parsecs​:

 $$\dfrac{4.82 \times 10^{16} \, m}{3.1 \times 10^{16}} = 1.3 \,pc$$

The meteor is $$\underline{ 1.3 \ parsecs}$$ from the Earth.​

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Stellar parallax

Stellar parallax is a tool used to calculate the distance between the sun and a nearby star, and is the apparent shift of a star against more distant stars. 

You can see the effect of parallax yourself if you put your thumb in front of you, and close one eye at a time. Your thumb will appear to move, but the background will not! 

The parallax angle, which can be used to calculate the distance, is calculated by measuring the apparent distance to a star from the Earth at a specific time period, and using precise measurements to determine the parallax angle. The distance to a star in parsecs can then be calculated by:

​$$d = \dfrac{1}{p}$$

where $$d$$ is the distance to the star in parsecs​ and $$p$$ is the parallax angle.​

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Example

A star close to the Earth, 61 Cygni has a parallax angle of $$0.29 \ arcseconds$$, calculate the distance from Earth to 61 Cygni in parsecs​.

State variables:

​$$p = 0.29 \ arcseconds$$

State equation:

$$d = \dfrac{1}{p}$$

Substitute values in and solve:

$$d = \dfrac{1}{0.29} \newline \\[0.1in]d = 3.5 \ pc$$

The star 61 Cygni is $$\underline{3.5 \ pc}$$ from the Earth. ​

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