Star Sizes
Published on May 19, 2006 at 6:31 pm.
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Filed under stars.
How big are stars? Well, they come in all sizes. Compared with Earth, they are all very big. But, the size of a star does not depend only on the mass of a star. For example, a star with half the mass of the Sun won’t be half the size of the Sun. In fact, depending upon what stage the star is in its life, it can be hundreds of times the size of our Sun. When a star begins to die, the core compresses and becomes hotter, and this makes the outer portions of the star expand and cool. The result is what we call a red giant star. The biggest stars are truly huge. Some red supergiants can be so big that many millions of stars the size of the Sun would fit inside them, and yet only dozens of times the mass of the Sun. That makes them mostly a very thin gas! The smallest stars are on the order of the size of Jupiter, the largest planet in our solar system. They may be the size of Jupiter, but they would have a hundred times the mass of Jupiter.
So, how do we measure the size of a star? That is tough. It is tough enough to measure the size of a planet, and you can see them as disks in a telescope! To measure the size of a planet, you first need to know how far the planet is from Earth. Then, you look through a telescope at the planet. The size of the planet can then be determined by how big the planet appears in the telescope. You measure the angular size of the planet, as it appears in the telescope. The actual diameter of the planet will be given multiplying the distance of the planet times the tangent of its angular size. In principle, this would work with stars, too. However, stars are much farther away than planets. The problem is that at the great distances of stars, they appear very tiny in the sky. In fact, the diffraction of light passing through a telescope blurs the image of the star so much that the blur is larger than the image of the star! This makes it impossible to determine the actual size of the star. Only a couple of stars are so large that they can be imaged as anything but a dot. The size of those stars can thus be directly measured. The sizes of the rest of the stars can only be determined with quite a bit more work.
The first thing in determining the sizes of stars is to realize that they shine because they are hot. There is a relationship in thermodynamics, called the Stefan-Boltzmann Law, that relates how bright an object shines to its surface area and to its temperature. If we assume that stars are spherical, then this works out very easily. As it turns out, stars are not perfect spheres, but the approximation is good for most stars. A few, though, such as Regulus, are decidedly NOT spheres (they are oblate spheroids), and so a different approach is needed. But, let’s keep it simple. For spherical stars, you can do a bit of algebra on the Stefan-Boltzmann law to find that the radius of the star (measured in solar radii) is approximately equal to square root of the luminosity of the star (measured in solar luminosities) divided by the temperature squared (measured in solar temperatures). Huh? What does this mean? Well, it turns out that the equations get simplest if you do comparisons rather than outright computations. So, we often compare values to those of the Sun, since the Sun is the nearest star to us and the easiest to measure. So luminosity measured in solar luminosities means how many times brighter than the Sun. A star of 12.3 solar luminosities is 12.3 times brighter than the Sun, and a star of 0.25 solar luminosities is ¼ as bright as the Sun. Likewise, solar temperatures are relative to the Sun’s temperature. A temperature of 1.5 solar temperatures means 50% hotter than the Sun (measured in Kelvin). A temperature of 0.82 solar temperatures means that the Kelvin temperature of the star is 82% that of the Sun. And, of course, solar radii are size measurements relative to the radius of the Sun.
OK, so we have a neat little formula that tells us how big the star is, compared to the Sun, as long as we know how bright it is and how hot it is. But, how do we find these measurements?
To find the temperature, you look at the light from the star. The hotter the star, the more blue it appears. The cooler the star, the more red it appears. In particular, you can determine the temperature of the star by measuring what color of light the star gives off most intensely and then using something called Wien’s Law. Wien’s law tells us that the temperature of a star, measured in Kelvin, is equal to 0.0029 divided by the wavelength of the most intense color of light radiated (measured in meters). Then, to find the temperature in solar temperatures, divide this value by 5800K (the temperature of the Sun).
Finding the brightness of the star is a bit trickier. First, you have to find the absolute magnitude of the star. To do that, you need to know the distance of the star. Finding the distance of a star can be tricky. For fairly nearby stars, it isn’t too bad. We use parallax. First you determine very accurately the position of the star. Then, six months later, Earth has moved to the other side of the Sun. If the star is near enough, then it appears to shift slightly in the sky due to the motion of the Earth. You measure the position very carefully again. You then compute the angular shift of the star in arcseconds. An arcsecond is 1/3600 of a degree. Half of this angular shift is the parallax angle. The distance of the star in units of parsecs is equal to the reciprocal of the parallax angle measured in arcseconds. Great! Now we have the distance. Next, to determine the absolute magnitude, you measure the apparent magnitude (how bright the star appears in the sky). The absolute magnitude is equal to the apparent magnitude plus 5 minus 5 times the logarithm of the distance in parsecs. That is: M = m + 5 – 5log(d).
OK, we have the absolute magnitude. Now what? Next, you compare the absolute magnitude of the star to the absolute magnitude of the Sun: 4.85. The brightness of the star in terms of solar luminosities is equal to 2.512 raised to the power (4.85 minus the absolute magnitude of the star). Now, you are ready to go back the first equation that I gave, since you now have the brightness of the star in solar luminosities and the temperature in solar temperatures.
So, that’s a bit of work to find the size of a star! Still, that is basically what you need to do. Astronomy is, in some ways, a very difficult science. We don’t always get to directly measure what we want to measure. We sometimes have to make all sorts of other measurements and then piece them together like a jigsaw puzzle to get what we want. Of course, I think that is fun!
-Astroprof
