5.41 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
  • p = parallax angle from earth to the nearby star
  • d = 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

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