# Kirchhoff’s Laws and Spectroscopy

FROM THE LECTURE SERIES: INTRODUCTION TO ASTROPHYSICS

## One of the most important things we can do with a telescope is spectroscopy. Historically, the advent of spectroscopy in the late 19th century served such a major motivation that it led to the invention of the new term—astrophysics. Spectroscopy follows some basic rules. These are known as Kirchhoff’s laws. Let’s learn more about these.

### The Laws

The number one Kirchhoff’s law is that optically thick sources emit a continuous Planck spectrum. That’s because the density is high enough for the photons to reach thermal equilibrium with the atoms. Kirchhoff’s law number two is that optically thin sources produce emission lines, like the Orion Nebula. Photons don’t interact with atoms on their way out, so there’s no way for them to reach equilibrium.

And, number three, if one has a hot, optically thick source, surrounded by a cooler, optically thin layer, that’s when they get an absorption spectrum.

### Getting the Absorption Spectrum

This third Kirchhoff’s law is the most interesting case, and requires some more explanation. As we know, stars are hotter and denser on the inside than the outside. So, a star is an example of a hot, optically thick source—the stellar interior—surrounded by a cooler, optically thin layer—the photosphere. The crucial point is that the mean free path is not the same for all photons. It depends strongly on the wavelength. Photons with a wavelength that’s just right to be absorbed by the surrounding atoms have a much shorter mean free path than photons with some random wavelength.

For example, in a star with a photosphere at 10,000° Kelvin, the mean free path at 0.656 microns is much shorter than at 0.6 or 0.7 microns. Photons with that specific red color can’t propagate as far as other photons. When we point our telescope at the star, we see down to an optical depth of 2/3, but the actual depth, in kilometers at which the optical depth is 2/3 depends on wavelength. At 0.656 microns, the star is more opaque, and we can’t see as far down into the interior.

This article comes directly from content in the video series Introduction to AstrophysicsWatch it now, on Wondrium.

### The Dark Lines

This implies that the photons of that wavelength that reach our telescope originated in a shallower, cooler layer of the star. And the blackbody radiation from that cooler gas is less intense than the radiation from deeper down. So, at 0.656 microns, the star appears fainter, and darker, than it does at adjacent wavelengths. We get a dark line in the spectrum.

If we could somehow resolve the stellar disk, and if we had special glasses to view the star at any wavelength we want, when we look at 0.656 microns the star would look fainter, and slightly bigger, by maybe a 1000 kilometers because the photosphere isn’t as deep. It occurs at a higher elevation from the center of the star.

This clearly elucidates why it’s a little too simplistic to say the dark lines come from absorption. It’s true those photons are getting absorbed, but they’re getting emitted at the same rate. It’s more accurate to say the dark lines are there because the star is more opaque at those wavelengths. We can’t see as far down into the stellar fog.

### Limb Darkening

There is also another interesting phenomenon. An image of the Sun with an appropriate telescope shows that it gets fainter near the rim of the circle; the so called ‘limb’ of the solar disk. Incredibly, this is not an optical illusion. The radiation from the limb really is less intense than the radiation from the center of the disc. This phenomenon is called limb darkening.

Our first reaction could be that that’s just because the Sun is a sphere and not a flat circle. What we’re seeing is the shadowing effect that one always gets when they shine a light at a sphere. But, we’re not shining a light at the Sun. The Sun’s shining at us. It’s glowing from within, sending power equally in all directions.

It’s not the spherical shape of the Sun that causes limb darkening. In fact, one can prove that a uniformly bright sphere doesn’t show any limb darkening. From any angle, it looks like a uniformly bright circular disk. This can be demonstrated using a ‘luminous decorative sphere’. People use them to spruce up their backyards and gardens. They don’t show limb darkening.

### Our Line of Sight and the Optical Depth

What causes limb darkening is that the Sun, like all stars, is a gaseous sphere, hotter and denser on the inside than the outside. When we look toward the center of the Sun, our line of sight pierces the Sun nearly perpendicular to the photosphere. The photons that come to us from that direction originated at an optical depth of about 2/3, corresponding to a certain depth in kilometres within the Sun.

But when we look toward the limb, our line of sight skims the surface. If we follow that grazing line of sight, we reach an optical depth of 2/3 at a higher altitude, further from the center of the Sun, and at higher altitude, the temperature is cooler, and the radiation is less intense. That’s why we get limb darkening.

It’s an interesting effect, and, it’s a useful way to test models for the Sun’s photosphere. By observing limb darkening in different colors, we can learn about how temperature increases with depth, and how the overall opacity of the Sun varies with wavelength.

### Common Questions about Kirchhoff’s Laws and Spectroscopy

Q: What does the first irchhoff’s law state?

The first Kirchhoff’s law is that optically thick sources emit a continuous Planck spectrum. That’s because the density is high enough for the photons to reach thermal equilibrium with the atoms.

Q: What is observed at the ‘limb’ of the solar disk?

An image of the Sun with an appropriate telescope shows that it gets fainter near the rim of the circle; the so called ‘limb’ of the solar disk.

Q: What causes limb darkening?

What causes limb darkening is that the Sun, like all stars, is a gaseous sphere, hotter and denser on the inside than the outside.