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

DESCRIPTION	EQUATION
Distance to a star in parsecs	$d = \dfrac{1}{p}$

Constants

CONSTANT	SYMBOL	VALUE
$1 \space astronomical \space unit$	$d$	$1.50 × 10^{11} \,m$
$1 \space light-year$	$d$	$9.46 × 10^{15} \, m$
$1 \space parsec$	$d$	$2.06 × 10^{5} \,AU \newline 3.08 × 10^{16} \,m \\[0.025in] \newline 3.26\, ly$

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.

Physics; Cosmology; KS5 Year 12; Units for astronomical distances

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 = $5.35 \,ly$

Convert into metres:

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

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:

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

Physics; Cosmology; KS5 Year 12; Units for astronomical distances

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.

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!

Physics; Cosmology; KS5 Year 12; Units for astronomical distances

Physics; Cosmology; KS5 Year 12; Units for astronomical distances

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.

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.