How do observations of he sun's surface tell us about conditions in the solar interior?

They do not tell us about the interior directly. It is possible to determine the temperature and pressure of the surface from the spectrum. After that, one has to resort to modeling of heat flow, pressure balance, nuclear reaction rate and knowledge of the sun's mass (from planet speeds)to come up with conditions at the center.

The pioneer of this process was S. Chandrasekhar, who won the 1983 Nobel Prize for his work that he had done about 40 years earlier. The Nobel Prize committee is often late to recognize great work.

There is a newer technique (developed in the last 20 years) call helioseismology that can also be used. It measures oscillations and modal patterns in the local in-an-out "drum-head like" vibrations of the solar surface to diagnose the variation of density with location inside. Precise Doppler shifts of atomic lines are used for the velocities. The math is quite complex.

For more about it, see
http://soi.stanford.edu/results/heliowhat.html

Observations of the Sun's surface, also known as the photosphere, provide valuable insights into the conditions in the solar interior. Here's how:

1. Spectroscopy: By analyzing the sunlight received from the Sun, scientists can perform spectroscopy. Spectroscopy involves studying the different wavelengths of light emitted or absorbed by different elements. By examining the spectra obtained from the Sun's surface, scientists can identify the presence of various elements like hydrogen, helium, and heavier ions. This information helps to understand the composition of the solar interior.

2. Solar Fluctuations: The Sun's surface displays various activities such as solar flares, sunspots, and prominences. These events are driven by the interactions of magnetic fields and plasma dynamics in the solar interior. By monitoring these fluctuations, scientists can study the behavior and properties of the magnetic fields within the Sun. This allows them to infer details about the processes occurring in the solar interior.

3. Helioseismology: Helioseismology is the study of solar vibrations or oscillations caused by sound waves bouncing around within the Sun. By observing these oscillations on the Sun's surface, scientists can extract valuable information about the conditions in the solar interior. The properties of these waves provide insights into the temperature, density, and composition of different layers within the Sun.

4. Solar Neutrinos: The Sun emits a vast number of subatomic particles called neutrinos. These particles are produced in nuclear fusion processes taking place in the Sun's core. By detecting and analyzing the neutrinos reaching the Earth's surface, scientists can measure their properties and use these measurements to probe the conditions in the Sun's interior, such as temperature and nuclear reactions.

In summary, observations of the Sun's surface through spectroscopy, monitoring solar fluctuations, analyzing helioseismic waves, and measuring solar neutrinos all contribute to our understanding of the conditions and processes taking place in the solar interior. These observations help to construct models and theories about the structure and evolution of our Sun.