Measuring stars
Stars appear as points of light, yet astronomers have measured their distances, brightnesses, temperatures, sizes, and masses.
Parallax
As Earth orbits the Sun, nearby stars appear to shift slightly against distant ones: parallax. The distance in parsecs is \(d = \frac{1}{p}\), where p is the parallax angle in arcseconds. One parsec is about 3.26 light-years. The European Space Agency's Gaia mission has measured parallaxes for over a billion stars.
d = \frac{1}{p}Luminosity and apparent brightness
Luminosity is the total energy a star emits per second; apparent brightness is how bright it looks from Earth, which depends on luminosity and distance by the inverse-square law. A very luminous distant star can look dimmer than a feeble nearby one. Knowing distance and apparent brightness gives luminosity.
The magnitude system
Astronomers often use magnitudes, from an ancient Greek system: brighter objects have smaller (even negative) numbers. A difference of 5 magnitudes is a factor of 100 in brightness. Sirius, the brightest night-time star, has an apparent magnitude of about -1.5; the faintest stars visible to the naked eye are about +6.
Temperature and colour
A star's colour reveals its surface temperature: red stars are coolest (about 3,000 K), yellow stars like the Sun are about 5,800 K, and blue stars are hottest (over 10,000 K). Spectral classes, from hottest to coolest, are O, B, A, F, G, K, M; the Sun is a G star. Annie Jump Cannon developed this classification in the early 1900s.
Sizes of stars
A star's luminosity depends on its size and temperature: \(L \propto R^2 T^4\). If two stars have the same temperature but one is 100 times more luminous, it must be 10 times larger. Stars range from white dwarfs about Earth-sized to supergiants hundreds or even over a thousand times the Sun's radius, such as Betelgeuse.
Binary stars and masses
Most stars are in binary or multiple systems. By observing two stars orbiting each other and applying Newton's version of Kepler's third law, astronomers calculate their masses. Stellar masses range from about 0.08 to over 100 solar masses; below about 0.08, objects cannot sustain hydrogen fusion and are called brown dwarfs.
The H-R diagram
The Hertzsprung-Russell (H-R) diagram plots stars' luminosity against temperature. About 90% of stars lie on a diagonal band called the main sequence, where they fuse hydrogen in their cores; along it, hotter stars are more luminous and more massive. Giants and supergiants lie above, and white dwarfs below the main sequence.
Worked example
A star has a parallax of 0.1 arcseconds, so its distance is 1/0.1 = 10 parsecs, about 32.6 light-years. If it appears exactly as bright as a second star at 5 parsecs, it must be 2^2 = 4 times as luminous, since it is twice as far away and brightness falls with the square of distance.
Key takeaways
Parallax gives distance, d = 1/p in parsecs; one parsec is 3.26 light-years; Gaia measured over a billion stars. Luminosity is total output; apparent brightness depends on distance. Smaller magnitudes are brighter. Colour reveals temperature; spectral classes are OBAFGKM. Luminosity depends on radius squared and temperature to the fourth power. Binary orbits give stellar masses. The H-R diagram shows the main sequence, giants, and white dwarfs.
Flashcard
Apparent shift of nearby stars
Check your understanding
A star has a parallax of 0.25 arcseconds. How far away is it?