What kind of stars are found here?

A star in this region of the Hertzsprung-Russell diagram has a temperature of roughly 4,000 kelvin (4,000 K), a luminosity one hundred times fainter than that of the Sun (0.01 × LSolar), and a radius one-third that of the Sun (R = 0.3 × RSolar). This star lies along the Main Sequence, where most stars (including the Sun) are found. The low temperature indicates quite red colours.

Try to read the values of L, T, and R for yourself from the diagram. Do you estimate values for the luminosity, temperature, and size of the star similar to those listed above?

Hertzsprung-Russell Diagram. The x-axis is labeled 'Surface Temperature' (in units of kelvins) with high temperatures around 60,000 on the left and low temperatures around 3,000 on the right. The points representing stars which appear furthest to the left are drawn in blue, those in the middle temperature range are drawn in yellow and orange, and those furthest to the right are drawn in red. The y-axis is labeled Luminosity (in units of solar luminosity), with low luminosities around 0.0001 at the bottom and high luminosities around 200,000 at the top. A third parameter, Radius (in units of solar radii), is also labeled. Lines of constant radius extend from the upper-left to the lower-right, covering the whole space with a set of parallel lines. In the lower-left corner we find the line labeled 'R is equal to 0.001 solar radii'; successive lines are labeled 0.01, 0.1, 1, 10, 100, and 1,000 solar radii, with the line for 'R is equal to 1,000 solar radii' being located in the upper-right corner. A series of blue, yellow, and orange points scattered along the 'R is equal to 0.01 solar radii' line is made up of white dwarf stars. A large set of red (and a few yellow and orange) points made up of giant stars appears between the 1 and 1,000 solar radii lines. The 1 solar radii line extends from the upper-left corner to the lower-right corner. The Main Sequence (a curved sequence of blue, yellow, orange, and red points) mainly follows the 0.1 to 10 solar radii lines; at the high luminosity end it curves up to slightly higher luminosities and at the low luminosity end it curves down to slightly lower luminosities. In addition, there are three green lines on the figure which intersect around the point near to 4,000 kelvins and 0.01 solar luminosities. One points down to the x-axis, one points left to the y-axis, and one is parallel to the black lines of constant radius (being drawn at a radius of around 0.3 solar radii).

The basic properties of the stars which lie along the Main Sequence differ from each other in predictable ways. They are all clustered tightly around a central line which runs down the middle of the Hertzsprung-Russell diagram. Because of this, for any one luminosity the range of possible radii and temperatures for Main Sequence stars is extremely small. Similarly, for any one temperature the range of possible radii and luminosities for Main Sequence stars is also fairly small. (Only at temperatures below 4,000 K does the distribution of stars along the Main Sequence begin to broaden, with many star of different luminosities and radii being found with the same temperature).

The key to understanding this property is to realize that there are families (groups) of straight lines running along and across the Hertzsprung-Russell diagram, which define all the locations in the diagram where stars have a shared property.

How can we use this narrow distribution to estimate the radii of a Main Sequence star from the luminosity L alone?

Consider the case of a Main Sequence star, with a luminosity just over one-hundredth that of the Sun. We will use the Hertzsprung-Russell diagram to estimate its radius R. Following the horizontal green line draw over from 0.013 on the y-axis, we see that a few red-colored Main Sequence stars lie along it (we include all of the red-colored stars which lie just above and just below the green cross). These are all the Main Sequence stars which have a luminosity of 0.013 LSolar. These stars are all related to one another in physical size. As stated above, diagonal lines drawn on the diagram represent the locations of stars all of the same size. Main Sequence stars with a luminosity of 0.013 LSolar thus extend over only a small range in radius.

Examine the Hertzsprung-Russell yourself, to estimate which two diagonal lines would best bound the location of these Main Sequence stars. These stars all lie below the line showing the location of all stars with the same radius as the Sun, and they lie above the the line showing the location of all stars with a radius ten times smaller than that of the Sun. It is easy to see that all Main Sequence stars with a luminosity of 0.013 LSolar have a radius between one-tenth and one times that of the Sun – but we can do better!

To keep the diagram easy to read, we have drawn lines of constant radius at the radius of the Sun, at ten times the radius of the Sun, at one hundred times the radius of the Sun, and such, marking only powers of ten. However, we could as easily draw a line of constant radius at twice the radius of the Sun, or two-hundred times the radius of the Sun, or at any other size scale.

With your mind, fill in the region between the diagonal line at one solar radius and the diagonal line at one-tenth solar radii with additional lines, all parallel to the two found on the diagram. These lines will mark the location of stars which are nine-tenth as large as the Sun, eight-tenth times as large as the Sun, seven-tenth times as large as the Sun, and such, down to two-tenths (one-fifth) the size of the Sun. You can use your powers of observation to see that the Main Sequence stars with a luminosity of 0.013 LSolar will all lie above the line where stars are two-tenths as large as the Sun, and below the line where stars are half as large as the Sun.

We have accomplished a great deal with a small amount of information! Using only the luminosity of a star and the fact that it lies on the Main Sequence, we have limited its radius to between one-fifth and one-half that of the Sun.