# Stellar distances and magnitudes

> CIE A-Level Physics · 9702
> Source: https://www.owlsprep.com/study/cie-9702-u29-stellar-distances-and-magnitudes/

This module covers core methods for measuring distances to nearby stars, and the magnitude scale for quantifying stellar brightness. You will learn key equations and common exam question approaches for this high-frequency topic.

**Prerequisites:** Trigonometry for right-angled triangles; [Inverse square law for electromagnetic radiation](https://www.owlsprep.com/study/cie-9702-u18-thermal-energy-transfer/)

## Learning objectives

- Calculate stellar distances using the stellar parallax method
- Distinguish between apparent magnitude and absolute magnitude
- Apply the distance modulus equation to find distance, apparent or absolute magnitude
- Recall the definition of a parsec and convert between parsecs and light years

## Stellar Parallax Method

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).

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

*Notation:* p

*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. $$p = \frac{1.4}{2} = 0.7 \text{ arcsec}$$

> **tip**
>
> Examiners almost always give the total 6-month shift, not the parallax angle. Always halve the given shift value.

> **Exam tip:** Always state the unit (arcseconds) for parallax angle to get full marks

## Parsec Definition and Distance Calculation

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

**Parsec** — 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.

*Notation:* pc

*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. $$d = 4 \times 3.26 = 13.04 \text{ ly}$$

> **info**
>
> Parallax is only accurate for stars closer than ~100 pc. Distant stars have parallax angles too small to measure from Earth.

## Apparent vs Absolute Magnitude

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.

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

*Notation:* m

**Absolute Magnitude** — 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.

*Notation:* M

**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.

## Distance Modulus Equation

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. $$10^1 = \frac{d}{10} \implies d = 10 \times 10 = 100 \text{ pc}$$

**Exam command terms**

- **Distinguish between** — Clearly state the difference between two terms, don't just define each *(Distinguish between apparent and absolute magnitude)*

- **Show that** — Show every step of your calculation, starting from the base equation *(Show that the distance is approximately 100 pc)*

## Common pitfalls

- **Wrong:** Forgetting to halve the total 6-month shift to get the parallax angle
  - Why it fails: Examiners regularly give the total shift as a trick to test understanding
  - Correct: Always remember: parallax angle = half the total shift measured over 6 months
- **Wrong:** Using the formula $d = p$ instead of $d = 1/p$
  - Why it fails: Inverse proportionality is easy to mix up: smaller parallax means farther distance
  - Correct: Recall: 1 arcsec parallax = 1 parsec, so $d = 1/p$ by definition
- **Wrong:** Thinking higher magnitude means brighter star
  - Why it fails: The ancient magnitude system ranks 1 (brightest) to 6 (dimmest), so it is reverse ordered
  - Correct: Always remember: lower magnitude = brighter star
- **Wrong:** Using distance in light years in the distance modulus equation
  - Why it fails: The equation is defined for distance in parsecs, with 10 pc as the reference for absolute magnitude
  - Correct: Convert all distances to parsecs before substituting into the distance modulus equation
- **Wrong:** Claiming parallax can be used to find distances to distant galaxies
  - Why it fails: Parallax angles for objects beyond 100 pc are too small to measure accurately
  - Correct: Only use parallax for nearby stars; other methods like standard candles are used for distant objects

## Cheatsheet

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

## 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.

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