Most stars are unresolved points of light in the sky, even when viewed through the largest telescopes. However, a few are big enough, bright enough, and close enough to allow us to measure their sizes directly. In an optical technique known as speckle interferometry, many short-exposure images of a star, each too brief for Earth's turbulent atmosphere to smear it out into a seeing disk, are combined to make a high-resolution map of the star's surface. In some cases, the results are detailed enough to allow a few surface features to be distinguished (Figure 17.5). As adaptive optics techniques continue to improve (see Chapter 5), it is becoming possible to combine the individual images in real time, again allowing very-high-resolution stellar images to be made. 
(Sec. 5.3)
Figure 17.5 (a) The swollen star Betelgeuse (shown here in false color) is close enough for us to directly resolve its size, along with some surface features thought to be storms similar to those that occur on the Sun. Betelgeuse is such a huge star (300 times the size of the Sun) that its photosphere spans roughly the size of Mars's orbit. Most of the surface features discernible here are larger than the entire Sun. (The dark lines are drawn contours, not part of the star itself.) (b) An ultraviolet view of Betelgeuse, in false color, as seen by a European camera onboard the Hubble Space Telescope. The bright spot at the bottom right of the star is more than 10 times bigger than Earth and at least 2000 K hotter than the surrounding 3000 K photosphere. Most astronomers regard it as a storm of some sort, perhaps similar to those that occur during the active phase of our Sun.
By measuring a star's angular size and knowing its distance from the Sun, astronomers can determine its radius by simple geometry. Optical astronomers have directly measured the sizes of a few dozen stars in this way. However, most stars are too distant or too small for this to be possible. Their sizes must be inferred by other means.
Recall from Chapter 3 that the radiation emitted by any hot body is governed by Stefan's law, which states that the energy emitted per unit area per unit time by a hot body—the body's energy flux—increases as the fourth power of the temperature. (Sec. 3.4) To determine the star's luminosity, we must multiply by the star's surface area, which is proportional to the square of the stellar radius. (Sec. 16.1) Combining these proportionalities, we have
where 
is the symbol representing proportionality. This radius-luminosity-temperature relationship is important because it demonstrates that knowledge of a star's luminosity and temperature can yield an estimate of its radius—an indirect determination of stellar size.
Let's consider some examples to clarify these ideas. The star known as Mira (Omicron Ceti) has a surface temperature of about 3000 K and a luminosity of 1.6 
1029 W. Thus, its surface temperature is half, and its luminosity about 400 times, the corresponding quantities for our Sun. The radius—luminosity—temperature relationship, rearranged into the more convenient form
then implies that the star's radius is 
times that of our Sun. If our Sun were this large, its photosphere would extend as far as the orbit of Mercury. A star as large as Mira is known as a giant. More precisely, giants are stars having radii between 10 and 100 times that of the Sun. Even larger stars, ranging up to 1000 solar radii in size, are known as supergiants. Because the color of any 3000 K object is red, Mira is a red giant.
Now consider Sirius B—a faint companion to Sirius A, the brightest star in the night sky. Sirius B's surface temperature is roughly 24,000 K, four times that of the Sun. Its total luminosity is 1025 W, about 0.04 the solar value. Substituting these quantities into our equation, we obtain a radius of 
solar radii— roughly the size of Earth. Sirius B is much hotter but smaller and far less luminous than our Sun. Such a star is known as a dwarf. In astronomical parlance, the term dwarf refers to any star of radius comparable to or smaller than the Sun (including the Sun itself). Because any 24,000 K object glows white, Sirius B is an example of a white dwarf.
The radii of the vast majority of stars (mostly measured using the radius—luminosity—temperature relationship) range from less than 0.01 to over 100 times the radius of the Sun. Figure 17.6 illustrates the estimated sizes of a few well-known stars.
Figure 17.6 Star sizes vary greatly. Shown here are the estimated sizes of several well-known stars, including a few of those discussed in this chapter.
