physics

While trying to plot Hipparcos satellite data into observational and theoretical Hertzsprung-Russel diagrams I got bogged down in a photometric definitions jungle, which I try to untangle in this post. The working astronomer will know all of this by heart, but at least I needed this workout.

Stellar spectroscopy, the study and classification of spectra, was born early in the 19th century when the German scientist Joseph von Fraunhofer discovered dark lines in the spectrum of the Sun (see previous post). He later observed similar lines in the spectra of stars and noted that different stars had different patterns of lines. Following Fraunhofer, Angelo Secchi and others developed classification schemes in the 2nd half of the 19th century, which via the Draper system, Harvard system and MKK system led to today's MK system. We follow some of the history just for the fun of it, and to understand how the "O, B, A, F, G, K, M" alphabet came about.

Fraunhofer lines are a set of spectral absorption lines named after the German physicist Joseph von Fraunhofer (1787–1826). The lines were originally observed as dark features in the optical spectrum of the Sun by the English chemist William Hyde Wollaston in 1802. In 1814, Fraunhofer independently rediscovered the lines and began to systematically study and measure the wavelengths where these features are observed. He mapped over 570 lines, designating the principal features (lines) with the letters A through K and weaker lines with other letters. Modern observations of sunlight can detect many thousands of lines. About 45 years later Kirchhoff and Bunsen noticed that several Fraunhofer lines coincide with characteristic emission lines identified in the spectra of heated elements. It was correctly deduced that dark lines in the solar spectrum are caused by absorption by chemical elements in the solar atmosphere.

This is a follow-on from a previous post on tangent spaces. It is part of a series of posts that I write on the basics of differential or Riemannian geometry, providing the necessary background for reading some of the more advanced textbooks on general relativity. I will introduce the derivative of a smooth map between manifolds and work a little bit with the definition(s) to become familiar with the thicket of mathematical notation. Then I narrow in on the special case where \(f\) is a real-valued function, introduce the notions of dual space and covectors, and finally show the connection to the differential known from basic calculus.

This is a follow-on from a previous post on differentiable manifolds. It is part of a series of posts that I write on the basics of differential or Riemannian geometry, providing the necessary background for reading some of the more advanced textbooks on general relativity. The following posts will cover tensors, covariant derivatives and parallel transport. The presentation will be informal, focusing mostly on motivation, definitions and key facts, not on mathematical proofs.

This is the first post in short series that I write on the basics of differential or Riemannian geometry. Differential geometry is a key mathematical foundation of general relativity, and the terminology presented here is a prerequisite for reading some of the more advanced textbooks, such as the ones by Carroll, MTW, Wald, Straumann or Hawking/Ellis. Here I start with differentiable manifolds. The following posts will cover tangent spaces, tensors, covariant derivatives and parallel transport. The presentation will be informal, focusing mostly on motivation, definitions and key facts, not on mathematical proofs.