AS1001-SEA: Script

What is this?

Here I test what’s going on.

This is the script for the ’Stars and Elementary Astrophysics’ (SEA) lectures, which are part of AS1001. The script covers the skeleton of facts for this course, including important equations and numbers. It is not a replacement for taking notes in the lectures. The lectures will cover everything in this script, but with more illustrations, explanations, figures, animations, and examples. The slides from the lectures will not be on Moodle. Important terms and concepts are marked with italics. Important equations are numbered. The textbook for this course is Kutner’s ’Astronomy - A Physical Perspective’. Synopsis: SEA answers the basic question ’what is a star?’. This includes measuring distances to stars, and finding out how large, hot, and massive stars are. SEA also covers the essential toolkit for astronomers - telescopes and instruments - as well as the basics of electromagnetic radiation. SEA is the fundament for all other astronomy modules.

Introduction

Astronomy is: the study of the stars. But astronomy also covers planets, gas clouds, galaxies, black holes, pulsars, the Universe itself - everything in the material world except things on Earth. Astronomy needs physics, chemistry, mathematics (and perhaps biology?).

What is a star? To the eye, stars are points of light at the night sky. They are grouped in constellations. These are arbitrary groupings of stars - chance projections, no physical groups. Examples for well-known constellations are Orion, Ursa Major, Taurus, Cassiopeia.

The constellations that are visible change over the year (due to the rotation of the Earth around the Sun) and over the night (due to the rotation of the Earth).

Looking at the night sky shows: Stars have different brightnesses and colours. But what is a star really?

Distances to stars

Fundamental problem: A point of light at the sky could be a nearby candle or a very distant supernova. To find out what stars really are, we need a method to determine the distance that is independent of the brightness. The most important method in this context is the parallax.

Parallax

The parallax method is based on triangulation. The basic principle is explained in Fig \ref{fig1}. If we can measure the angle p and the baseline s in the triangle, we can infer the distance: \(\tan{(p)} = s/d\), i.e. \(d = s/\tan{(p)}\). For small \(p\), we can use the small angle approximation, \(\tan{(tp)} \sim p\). That means:

\[d = s/p \label{eq1}\]

In astronomy, the distance Earth-Sun is used as baseline. The parallax is the apparent motion of stars on the sky caused by the rotation of the Earth around the Sun. Fig \ref{fig2} illustrates the parallax motion of a star. The parallax of stars \(p\) is measured in arcseconds (arcsec). 1" = 1 arcsec = 1/3600th of one degree. This is a very small angle.

The parsec

A parsec is defined as the distance of an object that has a parallax of 1 arcsec. The semi-major axis of the Earth’s orbit is \(1.5 \cdot 10^{11}\)m (defined as 1Astronomical Unit). If the parallax is measured in arcsec, the distance \(d\) to the star from the Sun in units of parsec is simply:

\[d = 1/p \label{eq2}\]

What is 1pc in metric units? Start with Equ \eqref{eq1}. Use \(s = 1\,\mathrm{AU} = 1.5 \cdot 10^{11}\)m. 1pc distance means \(p = 1" = (1/3600)\) degrees. Convert this to radians and put in Equ \eqref{eq1}: \(d = 1.5 \cdot 10^{11} / 0.0000048 = 3.1 \cdot 10^{16}\)m.

The closest stars to the Sun are the triple system \(\alpha\)Centauri at a distance of 1.3pc (and a parallax of 0.75"). There are several hundred stars within 10pc, among them Sirius, Vega, Procyon, Altair, some of the brightest stars on the sky.

Measuring the parallax

How accurately can we measure positions of stars? Parallaxes are very small angles. First successful measurements in 1838-1839. Today, 0.05“ can be done with ground-based telescopes. The satellite Hipparcos (1989-1993) measured parallaxes for 100.000 stars with 0.001” accuracy. The satellite Gaia (2013+) will get parallaxes for one billion stars with 0.0001" accuracy - this is still only 1% of the stars in the Milky Way.

This means: The parallax method only covers our cosmic neighbourhood. We need other methods for objects at larger distances. There are many more methods to determine distances of stars, some will be discussed in other parts of SEA1001. But all are based on the parallax.

Brightness of stars

Flux and luminosity

The luminosity \(L\) is the total energy emitted by a star per seconds, i.e. it is measured in Joule/sec or Watts. On Earth we only receive a part of this energy. The flux \(f\) is the energy per second that an observer on Earth measures. Luminosity is what the source emits, flux is what the observer receives.

Inverse square law for the propagation of light

Flux and luminosity are related through a basic law that describes the propagation of light in space. The light from a star spreads out isotropically (i.e. the same amount in all directions) over the surface of a sphere (see Fig \ref{fig3}). The flux is therefore:

\[f = L / (4 \pi d^2) \label{eq3}\]

Example: Assume a detector measures \(f=1\) for a star. Now we move the detector further away. At twice the distance, the light from the star has spread out to cover four times the surface, i.e. the same detector would collect 1/4 of the light. The flux drops with the inverse square of the distance.

Equ \eqref{eq3} contains three quantities, flux, luminosity and distance. The flux can be measured on Earth. The inverse square law can be used to determine luminosities, if the distance is known (from the parallax). It can also be used to determine distances, if the luminosity of a star is known. This second aspect leads to the concept of standard candles.

Standard candles and cepheids

A standard candle is an astronomical object with a known brightness, i.e. we know in some way how much light it emits. From this and the measured flux the distance can be derived using the inverse square law.

Cepheids are one example for standard candles. These are supergiant stars which vary periodically in brightness due to pulsation. Their pulsation period and their luminosity are related - the longer the period the larger the luminosity (see Fig \ref{fig4}). From the period we can derive the luminosity. From luminosity and flux follows the distance via the inverse square law.

Units for the brightness of stars

The unit of the flux is Watts per square meter Wm\(^{-2}\). This is very small for astronomical objects. Often used instead is the unit Jansky which is defined as \(1\,\mathrm{Jy} = 10^{-26}\)Wm\(^{-2}\)Hz\(^{-1}\)

Flux is measured on a linear scale, i.e. a source of 10Jy is ten times brighter than a source of 1Jy. This is inconvenient in astronomy, more useful would be a logarithmic scale \(\log{(f)}\). This leads to the concept of magnitudes.

Magnitudes

The unit magnitudes is derived from a system first used by the Greek astronomer Hipparcox (2nd century BC). In his catalogue of stars, 1st magnitude are the brightest stars, 6th magnitude are the stars just visible for the human eye. This system has now been adopted and extended for modern astronomy. The relation between fluxes and magnitudes is:

\[m_1 - m_2 = -2.5 \log{(f_1 / f_2)} \label{eq4}\]

Or conversely:

\[f_1 / f_2 = 10^{(m_1 - m_2) / -2.5} \label{eq5}\]

This is a logarithmic system, but with a scaling factor of 2.5. This factor means that 5mag difference correspond to a factor of 100 in flux. The zeropoint for the magnitude scale is Vega at \(m = 0.0\).

Magnitudes and distances

Usually magnitudes are apparent magnitudes (donated with small letter \(m\)). They relate to the flux measured on Earth and depend (as the flux does) on the distance.

But we can also define a magnitude that relates to the luminosity and does not depend on distance. This is the absolute magnitude (donated with capital letter \(M\)). The absolute magnitude is the magnitude a star would have at a distance of 10pc. Substituting Equ \eqref{eq3} into Equ \eqref{eq4} gives \(m - M = -2.5 \log{ ((10 / d)^2) } = -2.5 \cdot -2 \log{(d/10)}\), i.e.:

\[m - M = -5 \log{(d/10\,\mathrm{pc})} \label{eq6}\]

The quantity \(m-M\) is called the distance modulus and is another unit for distances of stars. If the distance of a star is known, the apparent magnitude \(m\) can be measured and the absolute magnitude \(M\) can be derived.

Bolometric magnitudes

Magnitudes are usually defined for specific parts of the spectrum (e.g., \(m_V\) for the visual light). The ’bolometric magnitude’ \(M_{\mathrm{bol}}\) is defined as the magnitude corresponding to the total energy received from the star (i.e. the flux integrated over the full spectrum).

Binary stars

Binary stars are two stars in mutual gravitational interaction orbiting their common center of mass (see Fig \ref{fig5}). Higher order binaries like triples and quadruples exist as well. Most stars are born as multiples.

We distinguish:

  • Visual binaries: Two stars are seen separately.

  • Spectroscopic binaries: Stars are not seen separately, but the spectrum shows two set of lines moving in opposite directions due to the Doppler shift (see below).

  • Eclipsing binaries: Stars are not seen separately, but one star eclipses the other in regular intervals.