# 10.2 Parallax

## Determining Distance using Parallax

• The principle of parallax is based on how the position of an object appears to change as the position of the observer changes
• For example, when observing the scale on a metre ruler, looking at eye level gets a different reading to viewing from above or below the scale
• Stellar parallax can be used to measure the distance to nearby stars
• Stellar Parallax is defined as:

The apparent shifting in position of a nearby star against a background of distant stars when viewed from different positions of the Earth, during the Earth’s orbit about the Sun

• It involves observing how the position of a nearby stars changes over a period of time against a ﬁxed background of distant stars
• From the observer's position the distant stars do not appear to move
• However, the closer object does appear to move
• This difference creates the effect of stellar parallax

#### Using Stellar Parallax

• A nearby star is viewed from the Earth in January and again in July
• The observations are made six months apart to maximise the distance the Earth has moved from its starting position
• The Earth has completed half a full orbit and is at a different position in its orbit around the Sun
• The nearby star will appear in different positions against a backdrop of distant stars which will appear to not have moved
• This apparent movement of the nearby star is called the stellar parallax

#### Calculating Stellar Parallax

• Applying trigonometry to the parallax equation:
• 1 AU = radius of Earths orbit around the sun
• = parallax angle from earth to the nearby star
• = distance to the nearby star
• So, tan(p)
• For small angles, expressed in radians, tan(p)p, therefore: p
• If the distance to the nearby star is to be measured in parsec, then it can be shown that the relationship between the distance to a star from Earth and the angle of stellar parallax is given by

• Where:
• p = parallax (")
• d = the distance to the nearby star (pc)
• This equation is accurate for distances of up to 100 pc
• For distances larger than 100 pc the angles involved are so small they are hard to measure accurately

#### Worked example

The nearest star to Earth, Proxima Centauri, has a parallax of 0.768 seconds of arc.

Calculate the distance of Proxima Centauri from Earth

1.  in parsec
2.  in light–years

Part (a)

Step 1: List the known quantities

• Parallax, p = 0.768"

Step 2: State the parallax equation

Step 3: Rearrange and calculate the distance d

Part (b)

Step 1: State the conversion between parsecs and metres

• From the data booklet:

1 parsec ≈ 3.1 × 1016 m

Step 2: Convert 1.30 pc to m

1.30 pc = 1.30 × (3.1 × 1016) = 4.03 × 1016 m

Step 3: State the conversion between light–years and metres

• From the data booklet

1 light–year ≈ 9.5 × 1015 m

Step 4: Convert 4.03 × 1016 m into light–years

= 4.2 ly (to 2 s.f)

#### Exam Tip

It is important to recognise the simplified units for arc seconds and arc minutes:

arcseconds = "

arcminutes = '

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