###### By Joshua N. Winn, Princeton University

## To measure a star’s luminosity**—**its total power output**—**we can use the flux-luminosity relationship. First, we measure distance to the star, *d*, maybe with parallax. Then, we measure the flux, *F*, the power per unit area we detect with our telescope. Finally, we calculate the luminosity as 4*πd*^{2}** ×** *F*.

### Temperature of the Star

What about the effective temperature, the temperature of its photosphere? We get that from spectroscopy. We measure the strength of the absorption lines and place the star on the spectral sequence, which as we saw last time, is a temperature scale. So, that gives us *L* and *T*-effective.

Now, let’s compare *L *versus *T*-effective. We’ll use logarithmic axes and to get oriented, we’ll mark the location of the Sun, which has an effective temperature of about 5800 Kelvin, and a luminosity of one, that is, one solar luminosity.

Now, let’s plot the data for the 1000 brightest stars in the sky. The effective temperatures range from around 3000 to 30,000 Kelvin. And the luminosities range from around 1/3 to 100,000 times the Sun’s. The Sun is way below average, it’s one of the least luminous stars on the chart.

### Pattern of ‘Main Sequence’ and ‘Giants’

We also see some patterns. A lot of the data points follow a stripe from the lower left to the upper right. The upward slope means that stars with hotter photospheres are more luminous. That makes intuitive sense. But there’s also a bunch of points higher up on the left. Those are very luminous stars, but with relatively cool photospheres. What’s going on with those?

The diagonal stripe is called the main sequence. Meanwhile, the stars on the left are called giants. That name suggests these stars are big. The key is that stars are approximately blackbodies. They do have absorption lines and other deviations from a pure Planck spectrum, but in broad terms the blackbody approximation is fine. So, we can use the Stefan-Boltzmann law: the luminosity equals the total surface area, 4*πr*^{2}, times the blackbody flux, σ x T^{4}.

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### Why Are the Giants Bigger than the Sun?

And since we’re looking at a logarithmic chart, let’s take the log. The result is an equation that connects the location of a point in our chart with the radius of the star. Say, we’re observing a bunch of stars and they all have the same radius *R*. Then 2-log *R* would be a constant. So, if we plot log *L* versus log *T*-effective, we would get a straight line, with a slope of 4.

Here’s what such a line looks like, on our axes. The temperature axis spans a little more than a factor of 10, which means log-*T* spans one unit. A line with a slope of 4 would rise 4 units in log-*L* as it crosses the chart. Let’s return to the data, and draw a line of slope 4 that goes through the Sun, so that any star the same size as the Sun—even if it differs from the Sun in temperature, luminosity, mass, age, composition, whatever—will lie somewhere on that line.

If we increase *R* by a factor of 10, log-*R* increases one unit, and 2-log *R* increases by 2. That causes the line to shift upward by 2 units. So, all stars 10 times bigger than the Sun should lie somewhere along this line. Using the same logic, we can draw a line for stars 100 times bigger than the Sun. Now we see why the stars in the upper left are called giants. They’re 10 to 100 times bigger than the Sun!

### Hertzsprung-Russell diagram

We can also see that the stars on the main sequence don’t all have the same size. Stars at the faint, cool end are the size of the Sun, or smaller and stars on the luminous, hot end are up to 10 times bigger than the Sun. This kind of chart, luminosity versus temperature, is called a Hertzsprung-Russell diagram, or HR diagram, after the astronomers who first drew them in the early 1900s.

Following astronomical tradition, they plotted absolute magnitude instead of luminosity. And instead of effective temperatures, which they didn’t know, at that time, they plotted a color index. The absolute magnitude is a log scale for luminosity. It’s equal to the apparent magnitude minus 5 times the log of the distance over 10 parsecs. And a color index is the difference between 2 apparent magnitudes, measured through different filters.

### The Color Index

So, it’s a log scale for a flux ratio. For example, the *B* minus *V* color index is the apparent magnitude measured through a standard blue filter, minus the apparent magnitude measured through a so-called “visual” filter, which is centered in the middle of the visible range. And since it’s all referenced to Vega, the color index of Vega is zero. For the Sun, *B* minus *V* happens to be 0.66. The important thing is that the color index is a proxy for effective temperature—blue stars are hot, red stars are cool—and the color index is much easier to measure with a pair of colored filters for your camera.

A traditional HR diagram shows the absolute *V* magnitude against *B* minus *V*. It has the same features we saw earlier, but everything is flipped left to right. That’s because a low value of *B* minus *V* means the star is relatively bright in *B* compared to *V*: It’s blue. And a star with high *B* minus *V* is red. So, that means the horizontal axis goes from blue to red, hot to cool. It’s a backwards temperature scale.

These data points are for the 1000 brightest stars in the sky. They come from a space telescope called Hipparcos, that measured the distances and visual fluxes of about 100,000 stars.

### Common Questions about the Luminosity of the Stars

**Q: What is Hertzsprung-Russell diagram?**

The chart that focuses on luminosity versus temperature is called a Hertzsprung-Russell diagram, or HR diagram, after the astronomers who first drew them in the early 1900s.

**Q: What is Hipparcos?**

Hipparcos is a space telescope that measured the distances and visual fluxes of about 100,000 stars.

**Q: How do we measure a star’s luminosity?**

To measure a star’s luminosity, we can use the flux-luminosity relationship. First, we measure distance to the star, *d*, maybe with parallax. Then we measure the flux, *F*, the power per unit area we detect with our telescope. Finally, we calculate the luminosity as 4-*pi* *d*-squared times *F*.