The Hertzsprung-Russell Diagram

The Hertzsprung-Russell (HR) diagram is one of the most important tools in all of astrophysics. It plots stars by their luminosity and surface temperature, revealing that stars are not randomly distributed but fall into distinct groups that correspond to different stages of stellar evolution. Every exam series includes at least one HR diagram question.

Structure of the HR Diagram

The HR diagram has:

Vertical axis: Luminosity (in solar luminosities L☉, or in watts). Luminosity increases upward. The scale is logarithmic, spanning roughly 10⁻⁴ L☉ to 10⁶ L☉.
Horizontal axis: Surface temperature (in kelvins). Temperature decreases to the right (this is the convention — hottest on the left). The scale is also logarithmic.

Alternatively, the horizontal axis may show spectral class (O B A F G K M, from left to right) or colour index (blue on the left, red on the right).

Common exam mistake: Plotting temperature increasing to the right. The HR diagram convention is temperature decreasing to the right (hottest on the left). Always check the axis before answering questions.

The Main Sequence

The most prominent feature is the main sequence — a diagonal band running from the upper left (hot, luminous) to the lower right (cool, dim). About 90% of all observed stars lie on the main sequence.

Main sequence stars are in a state of hydrostatic equilibrium: the outward radiation pressure and gas pressure from hydrogen fusion in the core exactly balance the inward gravitational compression. As long as hydrogen fusion continues, the star remains on the main sequence.

Key properties of main sequence stars:

Position on MS	Spectral type	Temperature (K)	Mass (M☉)	Luminosity (L☉)	Lifetime (years)
Upper left	O	40,000	60	10⁶	~1 million
	B	20,000	10	10³–10⁴	~10 million
	A	10,000	2.5	20–80	~1 billion
Middle	G (Sun)	5,800	1.0	1.0	10 billion
Lower right	K	4,500	0.7	0.3	15 billion
	M	3,000	0.3	0.01	>100 billion

The mass-luminosity relation for main sequence stars is approximately:

$\frac{L}{L_\odot} \approx \left(\frac{M}{M_\odot}\right)^{3.5}$

A star 10 times the Sun's mass is roughly 10³·⁵ ≈ 3,160 times more luminous. This extreme sensitivity means massive stars burn through their fuel far faster despite having more of it.

flowchart LR
    subgraph MS["Main Sequence (90% of stars)"]
        direction TB
        A["O/B stars: hot, blue, massive, luminous\nShort lives (~millions of years)"]
        B["G stars (Sun): medium temperature\nLifetime ~10 billion years"]
        C["K/M stars: cool, red, small\nVery long lives (>100 billion years)"]
    end
    subgraph RG["Red Giants"]
        D["Cool but very luminous\nLarge radius (10–100 R☉)\nHelium fusion or shell burning"]
    end
    subgraph WD["White Dwarfs"]
        E["Hot but very dim\nTiny radius (~Earth-sized)\nNo fusion — cooling remnants"]
    end
    subgraph SG["Supergiants"]
        F["Extremely luminous\nHuge radius (100–1000 R☉)\nMost massive star remnants"]
    end

Red Giants and Red Supergiants

Red giants are found in the upper right of the HR diagram — cool (3,000–5,000 K) but very luminous (10–1,000 L☉). Since L = 4πr²σT⁴, a star that is cool but luminous must have a very large radius. Red giants typically have radii of 10–100 solar radii.

Red giants are evolved stars that have exhausted the hydrogen in their cores and expanded. Hydrogen fusion continues in a shell around an inert (or helium-fusing) core.

Red supergiants are even more extreme — luminosities of 10⁴–10⁶ L☉ and radii of hundreds of solar radii. Betelgeuse, visible as the red shoulder of Orion, is a red supergiant with a radius about 900 times the Sun's.

Worked Example: Radius of a Red Giant

A red giant has luminosity L = 400 L☉ and surface temperature T = 3,500 K. Calculate its radius in solar radii.

Using Stefan's law as a ratio:

$\frac{L_{\text{giant}}}{L_\odot} = \left(\frac{r_{\text{giant}}}{r_\odot}\right)^2 \times \left(\frac{T_{\text{giant}}}{T_\odot}\right)^4$

$400 = \left(\frac{r_{\text{giant}}}{r_\odot}\right)^2 \times \left(\frac{3500}{5800}\right)^4 = \left(\frac{r_{\text{giant}}}{r_\odot}\right)^2 \times 0.1326$

$\left(\frac{r_{\text{giant}}}{r_\odot}\right)^2 = \frac{400}{0.1326} = 3017$

$\frac{r_{\text{giant}}}{r_\odot} = \sqrt{3017} = 54.9$

The red giant has a radius about 55 times the Sun's — it would extend well beyond the orbit of Mercury.

White Dwarfs

White dwarfs are found in the lower left of the HR diagram — hot (8,000–40,000 K) but very dim (10⁻⁴–10⁻² L☉). A hot but dim star must be very small. White dwarfs are typically the size of Earth (about 0.01 R☉) but with a mass comparable to the Sun's.

White dwarfs are the remnant cores of low- and intermediate-mass stars after they have shed their outer layers. They no longer undergo fusion — they simply cool and fade over billions of years. Their high density (about 10⁹ kg m⁻³) is supported by electron degeneracy pressure.

Worked Example: Radius of Sirius B

The white dwarf Sirius B has luminosity L = 0.026 L☉ and surface temperature T = 25,200 K. Find its radius.

$0.026 = \left(\frac{r}{r_\odot}\right)^2 \times \left(\frac{25200}{5800}\right)^4 = \left(\frac{r}{r_\odot}\right)^2 \times 356.0$

$\left(\frac{r}{r_\odot}\right)^2 = \frac{0.026}{356.0} = 7.30 \times 10^{-5}$

$\frac{r}{r_\odot} = 0.00855$

So r = 0.00855 × 6.96 × 10⁸ = 5.95 × 10⁶ m — about 93% of Earth's radius. An entire solar mass compressed into an Earth-sized ball.

Using the HR Diagram to Determine Stellar Properties

The HR diagram, combined with Stefan's law, allows you to determine any one of luminosity, temperature, or radius if you know the other two.

Since L = 4πr²σT⁴, lines of constant radius are diagonal lines on the HR diagram (because both axes are logarithmic):

Stars on the same constant-radius line have the same physical size
Moving up and to the left means hotter and more luminous for a given size
Moving down and to the right means cooler and less luminous for a given size

This is how we can identify star types just from their position:

HR region	Temperature	Luminosity	Implied radius
Upper left	Very hot	Very high	Moderate (main sequence) or large (supergiant)
Upper right	Cool	Very high	Must be very large → giant/supergiant
Lower left	Hot	Very low	Must be very small → white dwarf
Lower right	Cool	Low	Small → low-mass main sequence

Reading the HR Diagram in Exams

Typical exam questions ask you to:

Identify the type of star from its position (main sequence, red giant, white dwarf, supergiant)
Compare radii of two stars using L = 4πr²σT⁴
Describe the evolutionary path of a star on the HR diagram
Explain why red giants are luminous despite being cool (large radius)
Explain why white dwarfs are dim despite being hot (tiny radius)

Exam tip: When comparing stars, always use Stefan's law as a ratio. This eliminates constants and gives clean answers. Set up L₁/L₂ = (r₁/r₂)² × (T₁/T₂)⁴.

The Instability Strip and Variable Stars

The HR diagram also contains an instability strip — a vertical band where stars pulsate, periodically expanding and contracting. The most important pulsating stars are Cepheid variables, whose pulsation period is directly related to their luminosity (the period-luminosity relation). This makes Cepheids invaluable as standard candles for measuring distances to other galaxies — a key tool in the discovery of the expanding universe.

Summary

The HR diagram plots luminosity (vertical, increasing up) vs temperature (horizontal, decreasing right)
The main sequence runs upper-left to lower-right; 90% of stars lie on it
Red giants: cool + luminous = large radius (upper right)
White dwarfs: hot + dim = tiny radius (lower left)
Supergiants: extremely luminous, very large radii (top of diagram)
Stefan's law as a ratio is the key tool: L₁/L₂ = (r₁/r₂)² × (T₁/T₂)⁴
The mass-luminosity relation: L ∝ M³·⁵ (main sequence only)
Cepheid variables have a period-luminosity relation used to measure cosmic distances

A-Level Deep Dive: Hertzsprung-Russell Diagram

Spec mapping

Edexcel 9PH0 specification Topic 12 — Space covers stellar classification by spectral type and surface temperature, the construction and interpretation of the Hertzsprung-Russell (H-R) diagram with luminosity (or absolute magnitude) plotted against temperature on a reversed horizontal axis, and the use of the H-R diagram as a tool for tracking stellar evolution (refer to the official specification document for exact wording). The H-R diagram is examined principally on Paper 2 (Advanced Physics II), where Topic 12 is assessed alongside the other Year-2 topics. It is also a synoptic anchor: questions frequently combine H-R interpretation with Topic 5 (black-body radiation, Wien's and Stefan's laws), Topic 11 (parallax and the use of distance modulus), and Topic 12 (stellar evolution endpoints). The Edexcel formula booklet provides Wien's and Stefan's laws but does not provide the H-R diagram itself — its main-sequence diagonal, giant branch and white-dwarf locus must be reproduced from memory.

Worked example with full mark scheme

Question (8 marks):

Star X has an absolute magnitude of $M = +1.5$ and is classified as spectral type A0 with a surface temperature of approximately $10\,000\text{ K}$ .

(a) Place star X on a sketch H-R diagram, labelling the axes correctly, and identify the evolutionary phase it occupies. (4)

(b) The Sun has $M_\odot = +4.83$ and $T_\odot \approx 5\,800\text{ K}$ . Using Stefan's law as a ratio, estimate the radius of star X relative to the Sun. (4)

Solution with mark scheme:

(a) Step 1 — axes. The horizontal axis is surface temperature $T$ , plotted with temperature decreasing left to right (hot O/B stars on the left, cool M stars on the right). The vertical axis is luminosity $L$ (often $L/L_\odot$ on a logarithmic scale), increasing upward; equivalently absolute magnitude $M$ with magnitudes decreasing upward (since smaller $M$ means brighter).

M1 — both axes labelled with the temperature axis reversed. A common error: drawing temperature increasing left to right. That single slip loses both M1s for the diagram.

Step 2 — locate star X. With $T \approx 10\,000\text{ K}$ (A0 spectral class) and $M = +1.5$ , star X sits on the main sequence — the diagonal band running from hot luminous O stars (top-left) to cool faint M stars (bottom-right). At $T = 10\,000\text{ K}$ the main sequence has $M \approx +1$ to $+2$ , so star X is consistent with a main-sequence A0 star.

A1 — point correctly placed on the main sequence at the A-type position.

A1 — evolutionary phase identified as main sequence (core hydrogen burning).

(b) Step 1 — luminosity ratio from absolute magnitudes.

Pogson's relation gives $\dfrac{L_X}{L_\odot} = 10^{(M_\odot - M_X)/2.5} = 10^{(4.83 - 1.5)/2.5} = 10^{1.332} \approx 21.5$ .

M1 — correct application of the magnitude-to-luminosity conversion.

Step 2 — apply Stefan's law as a ratio. $L = 4\pi r^2 \sigma T^4$ , so $\dfrac{L_X}{L_\odot} = \left(\dfrac{r_X}{r_\odot}\right)^2 \left(\dfrac{T_X}{T_\odot}\right)^4$ .

M1 — Stefan's law written as a ratio (this avoids handling $\sigma$ and the $4\pi$ explicitly).

Step 3 — solve for the radius ratio.

$\left(\dfrac{r_X}{r_\odot}\right)^2 = \dfrac{L_X/L_\odot}{(T_X/T_\odot)^4} = \dfrac{21.5}{(10000/5800)^4} = \dfrac{21.5}{(1.724)^4} = \dfrac{21.5}{8.84} \approx 2.43$ .

So $\dfrac{r_X}{r_\odot} \approx \sqrt{2.43} \approx 1.56$ .

A1 — radius ratio approximately 1.5–1.6 times the solar radius.

A1 — answer quoted with appropriate significant figures and a comment that this is consistent with a main-sequence A-type star (slightly larger and considerably hotter than the Sun).

Total: 8 marks (M4 A4).

Specimen question modelled on the Edexcel 9PH0 Paper 2 format

Question (6 marks): A distant star has a measured surface temperature of $3\,500\text{ K}$ from its peak emission wavelength, and its spectrum places it in the K/M region.

(a) Explain how a single point on the H-R diagram is, in principle, insufficient to determine which evolutionary phase the star occupies, and describe what additional information resolves the ambiguity. (3)

(b) The star is found to have a luminosity $L = 1500\,L_\odot$ . Using the H-R diagram, identify the evolutionary phase and justify your answer. (3)

Mark scheme decomposition by AO:

(a)

M1 (AO1.2) — at low temperature (3500 K), three regions of the H-R diagram coexist vertically: the cool end of the main sequence (low $L$ ), the red-giant branch (high $L$ ), and red supergiants (very high $L$ ).
M1 (AO2.1) — measuring the apparent magnitude and the distance (e.g. via parallax) yields the absolute magnitude and hence the luminosity, fixing the vertical position.
A1 (AO2.1) — alternatively, spectral line widths (luminosity class) distinguish dwarfs from giants because pressure broadening is greater in dense main-sequence atmospheres.

(b)

M1 (AO1.1) — at $T = 3\,500\text{ K}$ a luminosity of $1500\,L_\odot$ is far above the main sequence, which at this temperature has $L \approx 0.05\,L_\odot$ .
M1 (AO2.1) — placing this point on the H-R diagram puts it firmly on the red-giant branch.
A1 (AO3.1) — justification that this is consistent with a post-main-sequence star where the core has contracted and the envelope expanded, raising luminosity while lowering surface temperature.

Total: 6 marks split AO1 = 2, AO2 = 3, AO3 = 1. This is a typical Paper 2 Topic 12 question — heavy on AO2 interpretation rather than recall, with a single AO3 mark for the evolutionary justification at the end.

Synoptic links

Connects to:

Topic 12 — Stellar classification (OBAFGKM): the spectral types are ordered by decreasing surface temperature, mapping directly onto the horizontal axis of the H-R diagram. O is hottest ( $\sim 30\,000\text{ K}$ +) at the left, M coolest ( $\sim 2\,500\text{ K}$ ) at the right. The mnemonic "Oh Be A Fine Girl/Guy Kiss Me" encodes the H-R horizontal ordering.
Topic 5 — Black-body radiation: stars are well-modelled as black bodies. Wien's law $\lambda_{\max} T = 2.9 \times 10^{-3}\text{ m K}$ converts peak emission wavelength to surface temperature — the input to the horizontal axis. Stefan's law $L = 4\pi r^2 \sigma T^4$ links the vertical ( $L$ ) and horizontal ( $T$ ) axes through stellar radius, which is why lines of constant radius appear as diagonals on the H-R diagram.