Stellar Parallax

The principle of parallax is based on how the position of an object appears to change depending on where it is observed from
- When observing the volume of liquid in a measuring cylinder the parallax principle will result in the observer obtaining different values based on where they viewed the bottom of the meniscus from

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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 star changes over a period of time against a ﬁxed background of distant stars
- To an observer the position of distant stars does not change with time
If a nearby star is viewed from the Earth in January and again in July, when the Earth is at a different position in its orbit around the Sun, the 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

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The Parallax Equation

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) = $\frac{1 A U}{d}$
For small angles, expressed in radians, tan(p) ≈ p, therefore: p = $\frac{1 A U}{d}$
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

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

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

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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

in parsec
in light–years

Part (a)

Step 1: List the known quantities

- Parallax, p = 0.768"

Step 2: State the parallax equation

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

Step 3: Rearrange and calculate the distance d

$d = \frac{1}{p} = \frac{1}{0.768} = 1.30 p c$

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

$\frac{4.03 \times 10^{16}}{9.5 \times 10^{15}}$ = 4.2 ly (to 2 s.f)

Examiner Tip

Make sure you know the units for arc seconds (") and arc minutes (')

1 arcminute is denoted by 1'
1 arcsecond is denoted by 1"

Stellar Parallax (OCR A Level Physics)

Revision Note