Stellar distances and magnitudes

CIE A-Level Physics· Unit 29: Astronomy and cosmology· 15 min read

1. Stellar Parallax Method★★☆☆☆⏱ 5 min

Stellar parallax is the apparent shift in position of a nearby star against the background of very distant fixed stars, caused by Earth's orbit around the Sun. The parallax angle is defined as half the total shift of the star observed over 6 months (one half of Earth's annual orbit).

📘 Definition

Parallax angle

p

The half-angle of the apparent shift of a star's position as Earth orbits the Sun, measured in arcseconds (arcsec).

Example:

A star that shifts 1.2 arcsec total between January and July has a parallax angle of 0.6 arcsec.

📐 Worked Example

A star is measured to have a total apparent shift of 1.4 arcsec over 6 months. What is its parallax angle?

1
Recall that the parallax angle equals half the total shift observed over 6 months:
2
$p = \frac{\text{Total shift}}{2}$
3
Substitute the given total shift:
4
$p = \frac{1.4}{2} = 0.7 \text{ arcsec}$

Exam tip:

Always state the unit (arcseconds) for parallax angle to get full marks

2. Parsec Definition and Distance Calculation★★☆☆☆⏱ 5 min

Using the small angle approximation for the right triangle formed by the Sun, Earth, and star, distance is inversely proportional to the parallax angle.

📘 Definition

Parsec

pc

A unit of stellar distance equal to the distance of a star with a parallax angle of 1 arcsecond. 1 parsec ≈ 3.26 light years.

Example:

A star 10 pc away has a parallax angle of 0.1 arcsec.

📐 Worked Example

Calculate the distance (in parsecs and light years) of a star with parallax angle 0.25 arcsec.

1
The relationship between distance $d$ (pc) and parallax $p$ (arcsec) is:
2
$d = \frac{1}{p}$
3
Substitute $p = 0.25$ arcsec:
4
$d = \frac{1}{0.25} = 4 \text{ pc}$
5
Convert to light years using 1 pc = 3.26 ly:
6
$d = 4 \times 3.26 = 13.04 \text{ ly}$

3. Apparent vs Absolute Magnitude★★★☆☆⏱ 5 min

The magnitude scale is a reverse logarithmic scale for stellar brightness. A lower magnitude value corresponds to a brighter star; a 5 magnitude difference corresponds to a 100× difference in brightness.

📘 Definition

Apparent Magnitude

m

The brightness of a star as observed from Earth. It depends on both the star's intrinsic luminosity and its distance from Earth.

📘 Definition

Absolute Magnitude

M

The apparent magnitude a star would have if it were placed at a distance of 10 parsecs from Earth. It measures intrinsic luminosity, independent of distance.

📐 Worked Example

Star X has m = 1.2, Star Y has m = 4.5. Which star appears brighter from Earth?

1
Recall the magnitude scale is reverse ordered: lower magnitude = brighter.
2
Since $1.2 < 4.5$ , Star X has a lower apparent magnitude, so it appears brighter from Earth.

4. Distance Modulus Equation★★★☆☆⏱ 5 min

The relationship between apparent magnitude $m$ , absolute magnitude $M$ , and distance $d$ (in parsecs) is given by the distance modulus equation, derived from the inverse square law for light.

m - M = 5 \log_{10} \left( \frac{d}{10} \right)

📐 Worked Example

A star has an apparent magnitude of 7.4 and absolute magnitude of 2.4. Calculate its distance in parsecs.

1
First calculate the distance modulus $m - M$ :
2
$m - M = 7.4 - 2.4 = 5$
3
Rearrange the distance modulus equation to solve for $d$ :
4
$5 = 5 \log_{10} \left( \frac{d}{10} \right) \implies 1 = \log_{10} \left( \frac{d}{10} \right)$
5
Exponentiate both sides with base 10:
6
$10^1 = \frac{d}{10} \implies d = 10 \times 10 = 100 \text{ pc}$

5. Common Pitfalls

Wrong move:

Forgetting to halve the total 6-month shift to get the parallax angle

Why:

Examiners regularly give the total shift as a trick to test understanding

Correct move:

Always remember: parallax angle = half the total shift measured over 6 months

Wrong move:

Using the formula $d = p$ instead of $d = 1/ p$

Why:

Inverse proportionality is easy to mix up: smaller parallax means farther distance

Correct move:

Recall: 1 arcsec parallax = 1 parsec, so $d = 1/ p$ by definition

Wrong move:

Thinking higher magnitude means brighter star

Why:

The ancient magnitude system ranks 1 (brightest) to 6 (dimmest), so it is reverse ordered

Correct move:

Always remember: lower magnitude = brighter star

Wrong move:

Using distance in light years in the distance modulus equation

Why:

The equation is defined for distance in parsecs, with 10 pc as the reference for absolute magnitude

Correct move:

Convert all distances to parsecs before substituting into the distance modulus equation

Wrong move:

Claiming parallax can be used to find distances to distant galaxies

Why:

Parallax angles for objects beyond 100 pc are too small to measure accurately

Correct move:

Only use parallax for nearby stars; other methods like standard candles are used for distant objects

6. Quick Reference Cheatsheet

Quantity	Symbol	Unit	Key Relationship
Parallax angle	$p$	arcseconds	$p = \frac{1}{2} \times 6 -month total shift$
Distance (parallax)	$d$	parsec (pc)	$d = \frac{1}{p}$
1 parsec		light years	$1 pc \approx 3.26 ly$
Apparent magnitude	$m$	dimensionless	Observed brightness from Earth
Absolute magnitude	$M$	dimensionless	Intrinsic brightness at 10 pc
Distance modulus	$m - M$	dimensionless	$m - M = 5 lo g_{10} (\frac{d}{10})$

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2022 · Paper 2
Parallax distance calculation
2023 · Paper 4
Distance modulus application
2021 · Paper 4
Distinguish magnitude types

Going deeper

syllabusCIE 9702 A-Level Physics SyllabusCheck page 56 for topic learning outcomes

What's Next

Stellar distances and magnitudes are the foundation of all astronomical distance measurement, which is required to map the universe and measure its expansion. After mastering this sub-topic, you can move on to more advanced methods for farther objects, such as standard candles and Hubble's law, which are core to understanding cosmology. Magnitude also connects directly to stellar classification and the Hertzsprung-Russell diagram, which describes the life cycle of stars. This topic is heavily assessed in Paper 4 of CIE A-Level Physics, so regular practice of calculations is critical for exam success.