The big idea: The Hertzsprung-Russell (H-R) diagram is a graph of every star's luminosity (how much power it radiates) against its surface temperature.
Where a star sits on it tells you what type of star it is, and lets you compare stars' temperatures, sizes and brightnesses at a glance.
One odd thing to remember: temperature runs backwards — hot stars are on the left, cool ones on the right.
[Diagram: phys-hr-diagram] - Available in full study mode
New words, plainly: Luminosity (L) = the total power a star radiates, in watts. (Not how bright it looks from Earth — that also depends on distance.)
Main sequence = the diagonal band where ordinary stars, fusing hydrogen, spend most of their lives. The Sun is on it.
Red giant / supergiant = a huge, cool, very luminous star (top-right). White dwarf = a tiny, hot, dim leftover core (bottom-left).
| Where it sits on the diagram | Star type | What that tells you |
|---|---|---|
| On the diagonal band through the middle | Main-sequence star (like the Sun) | Fusing hydrogen; hotter ones are brighter |
| Top-right (cool but very bright) | Red giant | Cool surface, but huge → still very luminous |
| Very top (extremely bright) | Supergiant | Enormous and the most luminous of all |
| Bottom-left (hot but very dim) | White dwarf | Hot surface, but tiny → very low luminosity |
How to read a position: Left ↔ right tells you the temperature (left = hotter).
Up ↔ down tells you the luminosity (up = brighter).
Off the main sequence? Top-right means big and cool (a giant); bottom-left means small and hot (a white dwarf).
A star's position fixes its size too. Two stars with the same luminosity but different temperatures must have different radii — the cooler one has to be bigger to radiate just as much power.
Hot star (left side)
- High surface temperature T
- Each square metre glows fiercely (L per area = σT⁴)
- For a given luminosity it can be small
Cool star (right side)
- Low surface temperature T
- Each square metre glows faintly
- To match the same luminosity it must be much bigger
- luminosity — total power the star radiates (W)
- Stefan-Boltzmann constant, 5.67 × 10⁻⁸ W m⁻² K⁻⁴ (given constant)
- surface area of the star, 4πr² for a sphere (m²)
- surface (absolute) temperature of the star (K)
- radius of the star (m)
Comparing to the Sun — the shortcut: Since L = 4πr²σT⁴, for two stars the σ and 4π cancel, so L ∝ r²T⁴.
Compare any star to the Sun and rearrange for the radius:
R_{star} / R_{sun} = (T_{sun} / T_{star})² × √(L_{star} / L_{sun})
Work in units of the Sun (L in Lsun, T in K) and you never need σ at all.
Worked example — radius of a red giant
A red giant has luminosity L = 8100 Lsun and surface temperature T = 2900 K. The Sun's surface temperature is 5800 K. How many times bigger than the Sun is this star? (Use Rstar/Rsun = (Tsun/Tstar)² × √(Lstar/Lsun).)
Solution
- Start from with . The constants cancel when comparing to the Sun, giving the radius ratio rule — write it first:
- Put in Tsun = 5800 K, Tstar = 2900 K and Lstar = 8100 Lsun:
- The Sun is twice as hot, so that bracket is 2² = 4; and √8100 = 90:
- Work it out:
Final answer
Rstar = (5800/2900)² × √8100 × Rsun = 4 × 90 = 360 Rsun. It is cool but enormous — a red giant.
Practice with real exam questions
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How this is tested: The H-R diagram appears on both papers:
- Paper 1A — identify / determine: read two stars' positions and compare their temperature, luminosity and radius; or calculate the ratio of two radii from their L and T using L ∝ r²T⁴. - Paper 2 — state / sketch: state a star's type from its position, or sketch where a star sits relative to the Sun, given its radius and temperature.
Classic trap: the temperature axis runs backwards (hot on the LEFT). And a star being far up (luminous) does not make it hot — red giants are cool but bright because they are huge.
IB-style question — ratio of two stellar radii
Two stars are plotted on an H-R diagram. Star X has luminosity 6400 Lsun and surface temperature 4350 K. Compared with the Sun (temperature 5800 K), determine the radius of Star X as a multiple of the Sun's radius. (Use Rstar/Rsun = (Tsun/Tstar)² × √(Lstar/Lsun).)
Solution
- The radius comes from with , which gives the radius ratio rule — write it first:
- Substitute Tsun = 5800 K, TX = 4350 K and LX = 6400 Lsun:
- Evaluate each piece: (5800/4350)² = 1.33² = 1.78, and √6400 = 80:
- Work it out — keep it as a multiple of Rsun:
Final answer
RX = (5800/4350)² × √6400 × Rsun ≈ 1.4 × 10² Rsun. Bright, a bit cooler than the Sun and much larger — a giant.
IB-style question — state the type and compare two stars
On an H-R diagram, Star P sits at the bottom-left (hot, very low luminosity) and Star Q sits at the top-right (cool, very high luminosity). (a) State the type of each star. (b) State which star has the larger radius, with a reason. [3]
Solution
- (a) Read the positions. Bottom-left = hot but dim; top-right = cool but bright:
Star P is a white dwarf; Star Q is a red giant (or supergiant). - (b) Compare the radii. Both come from L ∝ r²T⁴:
Star Q has a far higher luminosity yet a lower temperature, so its radius must be much larger to radiate that much power. - Answer the command term. State it plainly:
Star Q (the red giant) has the larger radius — it is cool but enormous, while the white dwarf is hot but tiny.
Final answer
(a) P = white dwarf, Q = red giant. (b) Q has the larger radius: it is more luminous but cooler, so by L ∝ r²T⁴ it must be much bigger.