Two of the most fundamental properties of any astronomical object are size and distance. How big is it, and how far away is it? The vast scales of cosmological distances are determined by a variety of methods, just as one uses different tools to measure the length of a piece of wood and the height of a building.

Clearly the distance of an object is tied to its apparent size. The distances to even relatively nearby stars reduces these behemoths to mere points of light, not spherical objects like our nearby star the Sun. So how do we know how big these objects are, or indeed how big any object in the sky is?

The most obvious is by direct imaging, taking a picture of the object, measuring its apparent size, and determining its actual size from our knowledge of its distance. Apparent size (angular size) is measured as an angle. For larger angles, your hand held at arm’s length provides a rough guide.

The relationship of distance, linear (actual) size, and angular (apparent) size is shown below.

Some simple math lets you determine any of these three values if you know the other two. Jupiter, for example, has an angular size of 49.6 arcseconds when it is at its closest to Earth, 3.97 astronomical units (au) away. [There are 360 degrees in a full circle. An “arcminute” is a smaller measure of angular size. There are 60 arcminutes in one degree. An arcsecond is even smaller. There are 60 arcseconds in one arcminute. To put this angular size into more familiar terms, the Moon is about 31 arcminutes in angular size. One astronomical unit is the average distance between the Sun and the Earth.] These two values let us calculate the equatorial diameter of Jupiter (its rapid rotation makes it bulge a little at the equator) as not quite 89,000 miles.

But Jupiter is large and quite close in astronomical terms, being a member of our solar system. Smaller and more distant solar system objects might need more clever means. Pluto is much smaller and appears only as a blurry smear in even the largest of Earth-based telescopes. Its largest moon Charon was discovered in 1978. Before the 2015 flyby of the Pluto-Charon system, the periodic mutual eclipses of Pluto and Charon provided a neat way to determine the diameters of both.

The light reflected from the two will be reduced when one passes in front of or behind the other. We know how fast Charon moves in its orbit around Pluto. By measuring how the total amount of light changes over time, we were able to get pretty accurate diameters way back in 1992. The “light curve” below shows how the amount of light dropped as Charon crossed in front of Pluto. It is modified from a figure in a paper by three astronomers describing measurements made in 1988. [Buie, M.W., Tholen, D.J., and Horne, K. 1992. Albedo Maps of Pluto and Charon: Initial Mutual Event Results. *Icarus ***97**, 211-227.]

While their purpose was more to create the best maps of the surfaces of these two bodies possible at the time than to measure their size, one could use the data to determine diameters, and they wouldn’t be more than 10% or less in error.

So how about stars? They are so far away (the nearest star is roughly 7,000 times more distant than Pluto) that even the largest appear only as points of light. Direct imaging won’t reveal size, and only a few stars will be properly aligned for the mutual eclipse method to work.

While stars are large and complex in their details, their large-scale properties are actually fairly simple. They are massive balls of hot plasma, with nuclear reactions at their cores counteracting the pull of gravity that would otherwise collapse them. They consist of hydrogen and helium with just a trace of anything else. The basic physics of such structures are well-known.

As much as I am tempted to derive the formula I’m about to give you—the habits of a retired chemistry and astronomy professor die hard—I’ll just give you the final result: the relationship among three stellar properties.

*L* is luminosity, where the luminosity of the Sun is 1. A star with twice the Sun’s luminosity would have a value of 2.

*R* is radius, where the radius of the Sun is 1.

*T *is the surface temperature of the star, in Kelvins.

*T _{s} *is the surface temperature of the Sun in Kelvins, 5772 K.

If we know how far away a star is and measure its apparent brightness, we can determine its luminosity, its actual brightness. The temperature of a star can be determined by measuring its spectrum, the way energy is distributed among different wavelengths of light.

As our last foray into math—Nothing more advanced than algebra, I promise!—let’s see how large the brightest star in our sky is. Sirius is most easily located by its proximity to the easily recognized winter constellation of Orion.

Sirius is bright mostly because it is relatively nearby—only 8.6 light years away, when the average distance between stars is about 5 light years. It does burn brighter and hotter than our Sun, with a luminosity 25.4 times that of the Sun, and a surface temperature of 9940 K.

I’ll even do the math for you. R for Sirius works out to 1.70 times the solar radius, or about 735,000 miles.

And finally, a series of images that show the relative sizes of several stars (and Jupiter), including our Sun. Enjoy!

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