H–R diagram explorer

A cool surface emits less energy per unit area than a hotter one, but a large star has much more area. This explorer lets you compare those effects without confusing a star’s total luminosity with how bright it appears from Earth.

Predict first. A model star has half the Sun’s effective temperature and twenty times its radius. Is it less luminous than the Sun, equally luminous, or more luminous?

Compare model stars

Temperature increases to the left on this conventional H–R diagram. Luminosity increases upward. Both axes are logarithmic: equal steps represent equal ratios, not equal additions.

Try a challenge

Can a cool star give out more light than a hot star?

Predict which matters more in this pair: temperature or emitting area. Load the hot, tiny star and the cool giant, then compare their luminosities in the plot and table.

Reveal the explanation

The 3,800 K model with radius 40 Suns radiates about 300 solar luminosities. The 15,000 K model with radius 0.012 Suns radiates only about 0.00657 solar luminosities. The much larger emitting area outweighs the lower temperature. These are constructed comparisons, not a track showing one star changing into another.

Explore · compare model stars, with the Sun as reference

The axes are logarithmic: equal vertical steps multiply luminosity rather than add to it. Hotter stars lie to the left. These dots are constructed comparisons, not measurements of individual stars or an evolutionary track.

Choose a comparison, then change it

Effective surface temperature: 5,772 KRadius / Sun’s radius: 1

The radius slider is logarithmic: moving the same distance multiplies the radius by the same factor. Its range is 0.01 to 100 solar radii.

Load the hot, tiny starLoad the cool giantDouble temperatureDouble radiusReset to the Sun

At 5,772 K and 1 solar radius, the model radiates 1 solar luminosity.

The static example works without JavaScript. If the controls do not activate, reload the laboratory or continue with the worked activity below.

We use L / L☉ = (R / R☉)² (T / 5772 K)⁴. Choosing temperature and radius freely does not guarantee a stable star. Moving this point is a comparison, not the life story of a star.

Read the same comparisons in a table

Constructed examples from the same temperature–radius model

Comparison	Temperature (K)	Radius (Sun = 1)	Luminosity (Sun = 1)
Sun reference	5,772	1	1
Cool, small star	3,200	0.2	3.78e-3
Hot, large star	20,000	5	3,600
Cool giant	3,800	40	301
Hot, tiny star	15,000	0.012	6.57e-3

Reference temperature: IAU 2015 Resolution B3. The exact teaching ratios use 5772 K as their solar reference; other entries are deliberately constructed, not catalog estimates. Download the model and comparison values.

Choose a comparison point from the selection menu, or change temperature and radius. The Sun is the reference. The other example dots are deliberately constructed comparisons, not measured properties of named stars.

Find the source of the extra light

Show the math and prediction answer

Preparation: ratios, squares and fourth powers. To calculate a fourth power, square a number and then square the result.

Effective temperature $T_{\rm eff}$ is defined through the star’s total emitted energy flux. For a spherical star with radius $R$,

$$L=4\pi R^2\sigma T_{\rm eff}^4.$$

The constant $\sigma$ is the Stefan–Boltzmann constant. For this teaching comparison we use the normalized relation

$$\frac{L}{L_\odot} =\left(\frac{R}{R_\odot}\right)^2 \left(\frac{T_{\rm eff}}{5772\ {\rm K}}\right)^4.$$

A radius twenty times larger gives $20^2=400$ times the area. Half the temperature gives $(1/2)^4=1/16$ times the emission per unit area. Together, the model has $400/16=25$ times the solar luminosity.

The reference temperature is the IAU nominal solar effective temperature. The displayed ratio is the lab’s defined normalized comparison; separately rounded nominal solar constants need not satisfy the dimensional formula with exact equality. IAU 2015, Resolution B3 (PDF)

At fixed radius, doubling temperature multiplies luminosity by 16. At fixed temperature, doubling radius multiplies luminosity by four. Use those predictions as checks before trying less familiar values.

A property diagram is not a movie of a star

Moving the controls does not evolve a star through time. Some combinations of radius and temperature may not describe a stable physical star at all. Actual evolutionary tracks require models of the interior, composition, mass, energy production, and mass loss.

The vertical quantity is luminosity: energy emitted per unit time. Apparent brightness also depends on distance and intervening absorption. The plot does not place nearby objects above distant ones simply because they look brighter.

Effective temperature is not the central temperature. It summarizes total surface emission even when a star’s spectrum is not a perfect blackbody at every wavelength.

Test your explanation

Can two stars at the same place in the temperature–luminosity diagram have different apparent brightnesses in our sky?

Answer

Yes. The diagram compares intrinsic temperature and luminosity. If otherwise identical stars are at different distances, their received fluxes differ. Dust can also change what reaches us. Placing a real star on an intrinsic H–R diagram requires enough information to account for those effects.

For educators

Preparation and objective. Compare ratios and read labeled axes; explain how radius and temperature jointly determine luminosity. In ten minutes, compare the cool large star with the solar reference using the static table: the area gain of 400 outweighs the flux reduction by 16, giving luminosity 25. For a twenty-minute extension, make two models with the same luminosity but different temperatures. Doubling the temperature requires one-quarter the radius to keep luminosity fixed, since $16(1/4)^2=1$. The calculations work on paper without controls. Ask learners to correct two misconceptions: dragging a point does not trace real stellar evolution, and a star’s effective surface temperature is not its core temperature.

Return to Why are some stars bright but cool? for the wider explanation.

References

• Blackhole.org. H–R comparison model, version stefan-boltzmann-ratios-1 (2026). Download model source; assumptions and reference checks.
• International Astronomical Union. “Resolution B3 on Recommended Nominal Conversion Constants for Selected Solar and Planetary Properties.” XXIX General Assembly, 2015. Official PDF.

The diagram, controls, and constructed examples are original teaching materials.