9.4: Diameters of Stars

Measuring the Size of a Star

It is easy to measure the diameter of the Sun. Its angular diameter—that is, its apparent size on the sky—is about 1/2°. If we know the angle the Sun takes up in the sky and how far away it is, we can calculate its true (linear) diameter, which is 1.39 million kilometers, or about 109 times the diameter of Earth.

Unfortunately, the Sun is the only star whose angular diameter is easily measured. All the other stars are so far away that they look like pinpoints of light through even the largest ground-based telescopes. (They often seem to be bigger, but that is merely distortion introduced by turbulence in Earth's atmosphere.) Luckily, there are several techniques that astronomers can use to estimate the sizes of stars.

Stars Blocked by the Moon

One technique, which gives very precise diameters but can be used for only a few stars, is to observe the dimming of light that occurs when the Moon passes in front of a star. What astronomers measure (with great precision) is the time required for the star's brightness to drop to zero as the edge of the Moon moves across the star's disk. Since we know how rapidly the Moon moves in its orbit around Earth, it is possible to calculate the angular diameter of the star. If the distance to the star is also known, we can calculate its diameter in kilometers. This method works only for fairly bright stars that happen to lie along the zodiac, where the Moon (or, much more rarely, a planet) can pass in front of them as seen from Earth.

Eclipsing Binary Stars

Accurate sizes for a large number of stars come from measurements of eclipsing binary star systems, and so we must make a brief detour from our main story to examine this type of star system. Some binary stars are lined up in such a way that, when viewed from Earth, each star passes in front of the other during every revolution. When one star blocks the light of the other, preventing it from reaching Earth, the brightness of the system decreases, and astronomers say that an eclipse has occurred.

Figure \(\PageIndex{1}\) is a chart of the light curve of an eclipsing binary and a diagram demonstrating the positions of the stars at each point on the curve. In this binary system, there is a larger, cooler red star, with a smaller, hotter blue star orbiting around the larger star. At position 1, the blue star is to the right of the red star. In this position, Earth observes light from both stars, so this is the maximum brightness of the system, represented by the brightness value of the chart at position 1. At position 2, the blue star has moved behind the red star. Although the blue star is small, it is very hot, so it is very bright. When the light from the blue star is blocked by the red star, the overall brightness seen at Earth drops by a large amount, which is why the brightness of the system is much lower at position 2. At position 3, the blue star is to the left of the red star and the brightness of the system is back to the position 1 value. Finally, at position 4, the blue star is in front of the red star. The blue star is blocking a small amount of light from the red star. This decrease in brightness is represented at position 4 on the chart. The drop in brightness at position 4 is significantly smaller than the large drop at position 2. Carefully tracking this curve reveals information about both stars.

Figure \(\PageIndex{1}\) : Light Curve of an Eclipsing Binary. As the smaller, hotter blue star passes behind and in front of the larger red star, the system's combined brightness dips sharply when the blue star is eclipsed and dips less when it blocks part of the red star's light. (CC BY 4.0; Fraknoi, et al. via Openstax Astronomy 2nd ed.) Accessible description of Figure \(\PageIndex{1}\).

The discovery of the first eclipsing binary helped solve a long-standing puzzle in astronomy. The star Algol, in the constellation of Perseus, changes its brightness in an odd but regular way. Normally, Algol is a fairly bright star, but at intervals of 2 days, 20 hours, 49 minutes, it fades to one-third of its regular brightness. After a few hours, it brightens to normal again. This effect is easily seen, even without a telescope, if you know what to look for.

In 1783, a young English astronomer named John Goodricke (1764-1786) made a careful study of Algol. Even though Goodricke could neither hear nor speak, he made a number of major discoveries in the 21 years of his brief life. He suggested that Algol's unusual brightness variations might be due to an invisible companion that regularly passes in front of the brighter star and blocks its light. Unfortunately, Goodricke had no way to test this idea, since it was not until about a century later that equipment became good enough to measure Algol's spectrum.

In 1889, the German astronomer Hermann Vogel (1841-1907) demonstrated that, like Mizar, Algol is a spectroscopic binary. The spectral lines of Algol were not observed to be double because the fainter star of the pair gives off too-little light compared with the brighter star for its lines to be conspicuous in the composite spectrum. Nevertheless, the periodic shifting back and forth of the brighter star's lines gave evidence that it was revolving about an unseen companion. (The lines of both components need not be visible for a star to be recognized as a spectroscopic binary.)

The discovery that Algol is a spectroscopic binary verified Goodricke's hypothesis. The plane in which the stars revolve is turned nearly edgewise to our line of sight, and each star passes in front of the other during every revolution. (The eclipse of the fainter star in the Algol system is not very noticeable because the part of it that is covered contributes little to the total light of the system. This second eclipse can, however, be detected by careful measurements.)

Any binary star produces eclipses if viewed from the proper direction, near the plane of its orbit, so that one star passes in front of the other (Figure \(\PageIndex{1}\)). But from our vantage point on Earth, only a few binary star systems are oriented in this way.

Diameters of Eclipsing Binary Stars

We now turn back to the main thread of our story to discuss how all this can be used to measure the sizes of stars. The technique involves making a light curve of an eclipsing binary, a graph that plots how the brightness changes with time. Let us consider a hypothetical binary system in which the stars are very different in size, like those illustrated in Figure \(\PageIndex{2}\). To make life easy, we will assume that the orbit is viewed exactly edge-on.

Even though we cannot see the two stars separately in such a system, the light curve can tell us what is happening. When the smaller star just starts to pass behind the larger star (a point we call first contact), the brightness begins to drop. The eclipse becomes total (the smaller star is completely hidden) at the point called second contact. At the end of the total eclipse (third contact), the smaller star begins to emerge. When the smaller star has reached last contact, the eclipse is completely over.

Figure \(\PageIndex{2}\) demonstrates how astronomers can measure the diameter of eclipsing binaries. Like the last example, this binary system is also a large red star with a smaller blue star orbiting around the larger star. Position 1 is first contact, where the blue star is just about to pass in front of the red star and position 2 is second contact where the eclipse is total. The shape of the light curve between these two positions is a drop in brightness. However, it is not a straight line down, because it takes time for the star to move from first to second contact. During the time interval between the first and second contacts, the smaller star has moved a distance equal to its own diameter. Positions 3 and 4 are third and last contact, respectively, and the light curve increase back to its normal value. During the time interval from the first to third contacts, the smaller star has moved a distance equal to the diameter of the larger star. If the spectral lines of both stars are visible in the spectrum of the binary, then the speed of the smaller star with respect to the larger one can be measured from the Doppler shift. Using the speed of the smaller star and the time between contacts, astronomers can calculate the diameters of both stars. The speed multiplied by the time interval from the first to second contact gives the diameter of the smaller star. We multiply the speed by the time between the first and third contacts to get the diameter of the larger star.

Figure \(\PageIndex{2}\) : Light Curve of an Edge-On Eclipsing Binary. As the smaller star crosses in front of the larger one, the time between first and second contact reveals the smaller star's diameter, and the time between first and last contact reveals the larger star's diameter. (CC BY 4.0; Fraknoi, et al. via Openstax Astronomy 2nd ed.) Accessible description of Figure \(\PageIndex{2}\).

In actuality, the situation with eclipsing binaries is often a bit more complicated: orbits are generally not seen exactly edge-on, and the light from each star may be only partially blocked by the other. Furthermore, binary star orbits, just like the orbits of the planets, are ellipses, not circles. However, all these effects can be sorted out from very careful measurements of the light curve.

Calculating Diameter with the Radiation Law

Another method for measuring star diameters makes use of the Stefan-Boltzmann law for the relationship between energy radiated and temperature. In this method, the energy flux (energy emitted per second per square meter by a blackbody, like the Sun) is given by

\[ F = \sigma T^4 \nonumber\]

where σ is a constant and T is the temperature. The surface area of a sphere (like a star) is given by

\[ A = 4\pi R^2 \nonumber\]

The luminosity (L) of a star is then given by its surface area in square meters times the energy flux:

\[ L = (A \times F) \nonumber\]

Previously, we determined the masses of the two stars in the Sirius binary system. Sirius gives off 8200 times more energy than its fainter companion star, although both stars have nearly identical temperatures. The extremely large difference in luminosity is due to the difference in radius, since the temperatures and hence the energy fluxes for the two stars are nearly the same. To determine the relative sizes of the two stars, we take the ratio of the corresponding luminosities:

\[ \frac{L_{\text{Sirius}}}{L_{\text{companion}}} = \frac{A_{\text{Sirius}} \times F_{\text{Sirius}}}{A_{\text{companion}} \times F_{\text{companion}}} = \frac{A_{\text{Sirius}}}{A_{\text{companion}}} = \frac{4\pi R^2_{\text{Sirius}}}{4\pi R^2_{\text{companion}}} = \frac{R^2_{\text{Sirius}}}{R^2_{\text{companion}}} \\ \frac{L_{\text{Sirius}}}{L_{\text{companion}}} = 8200 = \frac{R^2_{\text{Sirius}}}{R^2_{\text{companion}}} \nonumber\]

Therefore, the relative sizes of the two stars can be found by taking the square root of the relative luminosity. Since \( \sqrt{8200} = 91 \), the radius of Sirius is 91 times larger than the radius of its faint companion.

The method for determining the radius shown here requires both stars to be visible, which is not always the case.

Stellar Diameters

The results of many stellar size measurements over the years have shown that most nearby stars are roughly the size of the Sun, with typical diameters of a million kilometers or so. Faint stars, as we might have expected, are generally smaller than more luminous stars. However, there are some dramatic exceptions to this simple generalization.

A few of the very luminous stars, those that are also red (indicating relatively low surface temperatures), turn out to be truly enormous. These stars are called, appropriately enough, giant stars or supergiant stars. An example is Betelgeuse, the second brightest star in the constellation of Orion and one of the dozen brightest stars in our sky. Its diameter, remarkably, is greater than 10 AU (1.5 billion kilometers!), large enough to fill the entire inner solar system almost as far out as Jupiter. 

Further Exploration

• Use the Eclipsing Binary Simulator to explore how the eclipse timing can be used to determine the size of stars in a binary pair. Other traits can be explored as well, such as their mass, separation, and surface temperatures.
• Use the Stellar Luminosity Simulator to explore the relationship between a star's surface temperature, luminosity, and radius. Move the sliders to see what happens.
• Watch this star size comparison video for a striking visual that highlights the size of stars versus planets and the range of sizes among stars.