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Grade 10-11 (age 15-17)

high school Properties of Stars

Stellar Properties

Stars in the night sky differ in color and brightness, and a star's color is a clue to its surface temperature: blue stars are hot and red stars are comparatively cool. Plot stars on an H-R diagram, with surface temperature across and luminosity up, and most fall on the main sequence running from upper left to lower right, while giants, supergiants, and white dwarfs sit in separate places. Luminosity is set by radius and surface temperature through L = 4πR²σT⁴, so a star much larger than another at the same temperature is far brighter, and a hot star can still be dim if it is small. In the diagram, switching the group of stars shows where that group sits on the H-R diagram and, against the dashed lines of equal radius, how large its stars are.

Are all stars alike?

⭐ How stars differ
①Stars in the night sky differ in size, surface temperature, and brightness
②Hotter stars give off more short-wavelength light and look blue, while cooler stars look red
③Plot each star's surface temperature against its luminosity and the stars gather into a few groups; that plot is the H-R diagram

The H-R diagram (Hertzsprung–Russell diagram)

Main-sequence stars
💡 What to look for in the diagram
①Main-sequence stars gather on a band from the upper left (hot and bright) to the lower right (cool and dim), and about 90% of stars belong here
②Giants and supergiants sit at the upper right: they are cool but extremely bright because their radii are tens to a thousand times the Sun's
③White dwarfs sit at the lower left: they are hot but dim because they are only about the size of the Earth
④The dashed lines connect stars of equal radius, and radius grows toward the upper right

Physical properties of stars

Stefan–Boltzmann law
L = 4πR²σT⁴
Luminosity (L) is proportional to the square of the radius and the fourth power of the surface temperature. Doubling the temperature makes a star 16 times as luminous.
Wien's displacement law
λmax = b / T
The wavelength at which a star shines most strongly is inversely proportional to its surface temperature (b ≈ 2.9 × 10⁻³ m·K). The Sun peaks at about 500 nm.
🔬 Star color and surface temperature
①Blue stars: about 30,000 K and above (type O) and 10,000 to 30,000 K (type B)
②White stars: about 10,000 K (type A, Sirius A)
③Yellow stars: about 6000 K (type G, the Sun)
④Orange stars: about 4000 to 5000 K (type K, Aldebaran)
⑤Red stars: about 3500 K and below (type M, Betelgeuse, Proxima Centauri)

How bright stars are

Apparent and absolute magnitude
m − M = 5 log(d/10)
m is apparent magnitude, M is absolute magnitude, and d is distance in parsecs; the smaller the magnitude, the brighter the star.
📏 What magnitudes mean
①Apparent magnitude (m) is how bright a star looks from the Earth, so it depends on distance
②Absolute magnitude (M) is how bright the star would look from 10 pc away, so it compares how much light stars actually give off
③A difference of 1 magnitude is about 2.5 times in brightness, and a difference of 5 is exactly 100 times
④The larger m − M is, the farther away the star is

Worked Examples and Exam Practice

Example 1
Two stars differ by 5 magnitudes. How many times brighter is the brighter star?
1
Each step of 1 down in magnitude makes a star about 2.5 times brighter.
2
A difference of 5 magnitudes is defined as exactly 100 times in brightness (about 2.5⁵ ≈ 100).
▸ 100 times
The smaller the magnitude, the brighter the star. Every 5 magnitudes means exactly a factor of 100 in brightness.
Example 2
A star has the same surface temperature as the Sun and 100 times its radius. About how many times the Sun's luminosity does it have? (L = 4πR²σT⁴)
1
At the same temperature, the Stefan–Boltzmann law makes luminosity proportional to the square of the radius (L ∝ R²).
2
With 100 times the radius, the luminosity is 100² = 10,000 times.
▸ about 10,000 times
In the diagram, the 100 R⊙ dashed line crosses the Sun's temperature (about 5800 K) at a luminosity of 10⁴. That is why red giants are bright even though they are cool.
CSAT-style problem
Which statement about the H-R diagram and the physical properties of stars is correct?
①On the H-R diagram, surface temperature decreases toward the left
②Main-sequence stars run from the upper left to the lower right of the H-R diagram
③Red giants have high surface temperatures and small radii
④Absolute magnitude is a star's brightness at its actual distance
⑤The larger the magnitude number, the brighter the star
▸ ② Main-sequence stars run from the upper left to the lower right of the H-R diagram
1
Main-sequence stars are fusing hydrogen in their cores, and on the H-R diagram they form a band from the hot, bright upper left to the cool, dim lower right.
2
Temperature is higher toward the left (①), red giants are cool and have large radii (③), absolute magnitude is the brightness at 10 pc (④), and smaller magnitudes are brighter (⑤).

Summary

Properties of stars in brief
L = 4πR²σT⁴ / H-R diagram: across = surface temperature (hotter to the left), up = luminosity
A star's luminosity is set by its radius and surface temperature, and the H-R diagram shows the groups of stars and their stages of evolution.
🎯 Exam Points
①On the H-R diagram the horizontal axis is surface temperature (hotter to the left) and the vertical axis is luminosity
②The main sequence runs from upper left to lower right, and about 90% of stars belong to it
③Since L = 4πR²σT⁴, at the same temperature a larger radius means a brighter star
④Giants are bright despite being cool because they are large, and white dwarfs are dim despite being hot because they are small
⑤One magnitude means about 2.5 times in brightness, and five magnitudes mean 100 times
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