# How do I get temperature and diameter from a light curve?

I recently learned that short-period pulsating stars vary in diameter and temperature when they pulsate. If I have a light curve for such an object, is it possible to calculate the values of its diameter and temperature? If yes, how would I go about doing this?

Thanks!

You can definitely calculate the diameter of the star. You could use the radiation law given here:

Another method for measuring star diameters makes use of the Stefan-Boltzmann law for the relationship between energy radiated and temperature (see Radiation and Spectra). In this method, the energy flux (energy emitted per second per square meter by a blackbody, like the Sun) is given by

$$F=σT^4$$

where $$σ$$ is a constant and $$T$$ is the temperature. The surface area of a sphere (like a star) is given by

$$A=4πR^2$$

The luminosity ($$L$$) of a star is then given by its surface area in square meters times the energy flux:

$$L=(A×F)$$

Previously, we determined the masses of the two stars in the Sirius binary system. Sirius gives off 8200 times more energy than its fainter companion star, although both stars have nearly identical temperatures. The extremely large difference in luminosity is due to the difference in radius, since the temperatures and hence the energy fluxes for the two stars are nearly the same. To determine the relative sizes of the two stars, we take the ratio of the corresponding luminosities:

$$\frac{L_{Sirius}}{L_{companion}}=\frac{(A_{Sirius}×F_{Sirius})}{(A_{companion}×F_{companion})}=\frac{A_{Sirius}}{A_{companion}}=\frac{4πR^2_{Sirius}}{4πR^2_{companion}}=\frac{R^2_{Sirius}}{R^2_{companion}}$$

$$\frac{L_{Sirius}}{L_{companion}}=8200=\frac{R^2_{Sirius}}{R^2_{companion}}$$

Therefore, the relative sizes of the two stars can be found by taking the square root of the relative luminosity. Since $$\sqrt{8200}=91$$ , the radius of Sirius is 91 times larger than the radium of its faint companion.

The method for determining the radius shown here requires both stars be visible, which is not always the case.

This is just one way, there are quite a few ways that you can find online.

For temperature given here:

Measure the brightness of a star through two filters and compare the ratio of red to blue light. Compare to the spectra of computer models of stellar spectra of different temperature and develop an accurate color-temperature relation.  You can read this Astronomy question providing more information on how to find the temperature of a star.