What kind of stars are found here?

A star in this region of the Hertzsprung-Russell diagram has a temperature of roughly 12,000 kelvin (12,000 K), a luminosity 500 times less than that of the Sun (0.002 × LSolar), and a radius one hundred times smaller than the Sun (R = 0.01 × RSolar). This star lies along the narrow band in size, where white dwarf stars are found. The fairly low temperature indicates that this star is the end-product of a nova of a low-mass star that took place some time in the past (a younger star would not have had time to cool, and to move down the white dwarf sequence to such lower luminosities and cooler temperatures).

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 12,000 kelvins and 0.002 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 0.01 solar radii).

How can we find the temperature T of a white dwarf in this region of the Hertzsprung-Russell diagram, if we know its luminosity L?

We can use the Stefan-Boltzmann Law to relate the temperature (T), size (R), and luminosity (L) of a star to each other. Measuring L, R, and T in solar units, we say that:

Equation: L is equal to R-squared times T-raised to the fourth power

Our next step is to solve for T in this equation. Dividing through by R squared, we see that

Equation: T-raised to the fourth power is equal to L, divided by R-squared

We now take the fourth root of both sides of the equation, in order to find an expression for T. T is equal to the fourth root of L, divided by the square root of R.

Equation: The fourth root of T-raised to the fourth power is equal to T, which is equal to the fourth root of ( L, divided by R-squared ), which is equal to the fourth root of L, divided by the square root of R

Let us say that the luminosity of the star is exactly 0.00194 times that of the Sun, and the radius is roughly 0.01 in solar units, as the Hertzsprung-Russell diagram shows us that all white dwarfs are roughly one-hundredth of the size of the Sun. We can use these values to find the temperature, in units of the temperature of the Sun.

Equation: T is equal to the fourth root of 0.00194 divided by the square root of 0.01, which is equal to 2.10 in units of solar temperature (2.10 times hotter than the Sun).

This white dwarf is almost three times hotter than the Sun. Now, we can convert this value into kelvins, as we know that the Sun has a surface temperature of 5800 K.

Equation: T is equal to 2.10 in units of solar temperature, which is equal to 2.10 in units of solar temperature times 5,800 K divided by 1 in units of solar temperature, which is equal to 12,172 K.

We estimated a value of T = 12,000 K from the diagram, for stars found in this area – a reasonable value!