Solar astronomy is the scientific study of the star nearest us, our sun. The word 'solar' comes from the Latin word 'solaris'/'sol' ('sun'), and the word 'astronomy' comes from the Greek 'astron' (star) and 'nomos' (law). As nearly as we can tell, our sun appears to be a typical main sequence G2 V yellow dwarf star, one of the most common types of stars in the sky, so what we can learn from it may tell us much about the stars too far away for us to study so directly. In turn, what we find out may suggest types of stars to focus upon when we seek planets capable of sustaining our known form of life.
Solar astronomy is itself a specialisation of astronomy, but is increasingly attracting scientists from a wide range of specialisations outside the field of astronomy. Certain skills applicable to terrestrial specialisations can also be applied to inferences about the sun's structure. Some terrestrial directions of study, such as the aurora borealis, blur directly into solar astronomy by way of plama physics and electromagnetism.
What follows is a representative overview of some tools and approaches used by modern solar astronomy. It should not be considered an exhaustive list.
Ever since the 1957 paper "Synthesis of the Elements in Stars", it has been understood that the sun is fueled by nuclear reactions, and that most existing elements have been created by such nuclear reactions. Elements present in the sun can be measured through atomic emission spectroscopy. In turn, specific patterns of elements identify different types of atomic reactions.
Terrestrial seismologists have long known that shock waves, such as those produced by earthquakes, travel through different densities of materials at different speeds. Measuring changes in speed and amplitude through spectrograph Doppler shift gives insight into the geological structure of the world beneath our feet. In a similar fashion, helioseismologists measure pressure waves at the sun's surface to try to gain insight into the sun's internal structure. The Solar and Heliospheric Observatory (SOHO) uses sound waves travelling through the photosphere to derive internal structure.
Currents in the sun have different temperatures than their surrounding regions. Sunspots are cooler than their surroundings. The corona is immensely hotter than the surface of the sun. Subtleties of temperature differential can tell us what is happening just under the surface.
The sun's magnetic field dominates all other magnetic activity in the solar system. In the same way as magnetic anomalies on earth indicate hidden magnetic masses or extraterrestrial influence, irregularities and changes in the field can suggest relative movement of various parts of the inner sun. Solar magnetic turbulence affects the creation of sunspots and solar flares, so understanding and predicting changes in the sun's magnetic field is essential to maintaining a reliable network of communication satellites and expanding exploration of our solar system.
Burbidge E, Burbidge GR, Fowler WA, Hoyle F. (1957). Synthesis of the elements in stars. Rev Mod Phys 29(4):547-650.