18.3 Diameters of Stars

Measuring Star Diameters

Stars range enormously in size. Some are smaller than Earth, while others would engulf the orbit of Jupiter. But how do you measure the diameter of something so far away that it appears as a point of light, even through a telescope? Astronomers use several indirect methods, each suited to different situations.

Methods of Stellar Diameter Measurement

Lunar occultations happen when the Moon passes in front of a star from Earth's perspective, blocking the star's light. The star doesn't vanish instantly. Instead, it fades over a tiny fraction of a second as the Moon's edge crosses the star's disk. A star with a larger angular diameter takes slightly longer to fade out and reappear than a smaller one. By timing this precisely, astronomers can calculate the star's angular size.

Eclipsing binary systems are pairs of stars orbiting each other in a plane aligned with our line of sight, so one star periodically passes in front of the other. The shape and duration of each eclipse depend on the relative sizes of the two stars and how far apart they orbit. Algol (β Persei) is a classic example. By carefully analyzing the timing of eclipses, astronomers can work out the diameter of each star in the pair.

Interferometry combines light collected by multiple telescopes separated by large distances. This effectively creates a virtual telescope with much higher angular resolution than any single instrument could achieve. It allows astronomers to resolve the tiny angular diameters of nearby stars directly.

Light Curves and Stellar Size

A light curve is a graph of an eclipsing binary's brightness over time. It holds a surprising amount of information about the sizes of the two stars.

• Eclipse depth reflects the size difference between the stars. A deep dip means one star is much larger or brighter than the other (like in Algol). A shallow dip means the two stars are closer in size (like in W Ursae Majoris systems).
• Eclipse duration depends on both the stars' sizes and their orbital separation. Longer eclipses point to larger stars or a tighter orbit; shorter eclipses suggest smaller stars or a wider orbit.

By modeling the full shape of the light curve, astronomers can solve for the relative diameters of both stars in the system.

Estimating Star Sizes

The Stefan-Boltzmann Law

Even without a binary system or an occultation, astronomers can estimate a star's size using physics. The Stefan-Boltzmann law connects a star's luminosity ($L$), radius ($R$), and surface temperature ($T$):

$L = 4\pi R^2 \sigma T^4$

Here, $\sigma$ is the Stefan-Boltzmann constant. The key idea: luminosity depends on both how big the star is (the $R^2$ term) and how hot it is (the $T^4$ term). If you know $L$ and $T$ from observations, you can solve for $R$.

This leads to two useful rules of thumb:

• At the same temperature, a more luminous star must be larger. Rigel is far more luminous than Sirius, even though both are hot blue-white stars, so Rigel must have a much bigger radius.
• At the same luminosity, a cooler star must be larger. A cool star radiates less energy per square meter of surface, so it needs a bigger surface area to produce the same total luminosity.

Worked example: Suppose Star A has a surface temperature of 10,000 K and Star B has a surface temperature of 5,000 K, but both have the same luminosity.

1. Set their luminosities equal: $4\pi R_A^2 \sigma (10{,}000)^4 = 4\pi R_B^2 \sigma (5{,}000)^4$
2. Cancel the constants ($4\pi\sigma$) from both sides.
3. You get $R_B^2 / R_A^2 = (10{,}000)^4 / (5{,}000)^4 = 2^4 = 16$.
4. So $R_B / R_A = 4$. Star B must have 4 times the radius of Star A.

(Note: the ratio of radii is 4, not 16. The factor of 16 applies to $R^2$, not $R$ itself.)

Stellar Classification and Size

Spectral classification (O, B, A, F, G, K, M) tells you a star's surface temperature, and the luminosity class (I through V) tells you roughly how luminous it is. Together, these place a star on the H-R diagram, which immediately gives you a sense of its size.

• Main sequence stars (luminosity class V) follow a tight relationship: hotter main sequence stars are larger and more luminous. The Sun is a G2V star with a radius of about 696,000 km. A hot O-type main sequence star can be 10+ times that size, while a cool M-type red dwarf might be only a fraction of the Sun's radius.
• Giants and supergiants (luminosity classes I through III) are far larger than main sequence stars of the same temperature. That's exactly what the Stefan-Boltzmann law predicts: same temperature but much higher luminosity requires a much bigger radius.

To convert a star's angular diameter (measured through occultation, interferometry, or eclipsing binaries) into a true physical diameter, you need to know its distance. Stellar parallax is the most direct way to get that distance for nearby stars.