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.

Binaries are useful for two reasons: a) The orbits determined by gravitational forces, i.e. by studying them we can determine masses of stars. b) Eclipses can be used to determine the sizes of stars. Binaries are therefore important to determine fundamental properties of stars (i.e. what a star really is).

Doppler effect

The wavelength that an observer measures depends on the relative motion between the observer and the light source. If the source is moving towards the observer, the light is blueshifted (\(\lambda < \lambda_0\)). If the source is moving away from the observer, the light is redshifted (\(\lambda >\lambda_0\)). This is called the Doppler effect (analogous to the Doppler effect with sound waves).

The shift in wavelength due to the Doppler effect in the light of a star is:

\[\Delta\lambda / \lambda = v/c \label{eq7}\]

Here \(c\) is the speed of light, \(\Delta\lambda\) is the wavelength difference \((\lambda - \lambda_0)\), and \(v\) is the radial velocity of the star, i.e. the relative speed of the star along the line of sight of the observer. Equ \eqref{eq7} is only valid for \(v<<c\).

The Doppler effect shifts lines in the spectra of stars (see Sect. XXX). From these shifts and with Equ \eqref{eq7} the radial velocities of stars can be measured. The Doppler effect has many applications in astronomy; one example is the study of binary stars.

Orbits of binaries

Fig \ref{fig5} shows the orbit for a binary. We consider here circular orbits for simplicity. The orbital period is \(P = 2 \pi r / v\). In the binary system both periods are the same, \(P_1 = P_2\). From these two equations follows:

\[r_1 / v_1 = r_2 / v_2 \label{eq8}\]

With the definition of the center of mass, \(m_1 r_1 = m_2 r_2\), we obtain:

\[v_1 / v_2 = r_1 / r_2 = m_2 / m_1 \label{eq9}\]

This means, the more massive star is orbiting at a shorter distance from the center of mass and with a smaller velocity.

Kepler’s 3rd law for binaries

Newton’s law of gravity is: \(F = (G m_1 m_2) / (r_1 + r_2)^2\)

In a binary system, the gravitational force equals the force to keep the star in orbit:

\[m_1 v_1^2 / r_1 = G m_1 m_2 / (r_1 + r_2)^2 \label{eq10}\]

With \(v_1 = 2 \pi r_1 / P\) follows \(4 \pi^2 r_1 / P^2 = G m_2 (r_1 + r_2)^2\). With \(R = r_1 + r_2\) this gives after a few more steps (see textbook):

\[4 \pi^2 R^3 / G = (m_1 + m_2) P^2 \label{eq11}\]

In solar system units (distance between the stars in AU, period in years, masses in solar masses), this equation simplifies to:

\[R^3 = (m_1 + m_2) P^2 \label{eq12}\]

This is Kepler’s 3rd law.

In a spectroscopic system, the radial velocities \(v_1\) and \(v_2\) can be measured, together with the period. This immediately gives \(r_1\) and \(r_2\) and thus \(R\), as well as the ratio \(m_1\) and \(m_2\). With Kepler’s 3rd law, we can calculate the sum of the masses. Together this gives individual masss for the components.

Factors that are neglected here: elliptic orbits, inclination of orbits against the line of sight (for spectroscopic binaries) or the sky (for visual binaries).

Eclipsing binaries

Fig \ref{fig6} shows the lightcurve of an eclipsing binary system. From the lightcurve, \(t_1\), \(t_2\), \(t_3\), \(t_4\) can be measured. The geometry of the system then gives expressions for the diameter \(D_1\) and \(D_2\) of the stars:

\[(t_4 - t_1) / P = (D_1 + D_2) / (2 \pi R) \label{eq13}\]

\[(t_3 - t_2) / P = (D_1 - D_2) / (2 \pi R) \label{eq14}\]

With these two equations the sizes of the stars can be derived in units of the orbital separation \(R\). The radial velocities (\(v_1\), \(v_2\)) and the period \(P\) yield \(R\), and thus the radii of the stars. This only works in an eclipsing binary which is also a spectroscopic binary. This rare type of system is therefore extremely important in astronomy.

Radiation from stars

Electromagnetic radiation

Summary of important facts:

  • EMR travels with the speed of light (\(c = 3\cdot10^8\)ms\(^{-1}\)

  • EMR has wave-like properties (e.g., interference).

  • EMR is described by a wavelength \(\lambda\) or a frequency \(\nu\), with \(c = \nu\lambda\).

  • EMR also has particle-like properties (e.g., photoelectric effect).

  • A ’light particle’ or photon is a packet of energy with \(E = h \nu = h c / \lambda\) (\(h\): Planck’s constant).

Blackbodies

A blackbody is an idealised body in thermodynamical equilibrium with its surroundings, absorbing all radiation incident on it, and then re-radiating it. Every warm object – including stars – emits blackbody radiation caused by the thermal motions of its particles.

A blackbody emits some energy at all wavelengths. The spectrum of a blackbody – the emitted flux as a function of wavelength – is described by the Planck function shown in Fig. \ref{fig7} (in units of Wm\(^{-2}\)Hz\(^{-1}\)ster\(^{-1}\)).

\[B(\nu,T) = \frac{2h\nu^3}{c^2} \frac{1}{e^{\frac{h\nu}{kT}} - 1} \label{eq15}\]

At every wavelength, a hotter blackbody emits more energy than a cooler one. The peak of the Planck function shifts towards shorter wavelengths for higher temperatures. For a blackbody with temperature \(T\) there is a wavelength \(\lambda_{\mathrm{max}}\) at which is radiates its maximum energy, the peak of the Planck function. This is described by Wien’s law.

\[\lambda_{\mathrm{max}} T = 0.0029\,\mathrm{m K} \label{eq16}\]

Example: Sun with \(T = 5800\)K, i.e. \(\lambda_{\mathrm{max}} \approx 500\)nm

For long wavelengths and small frequencies (i.e. radio radiation), Planck’s law can be approximated by the Rayleigh-Jeans law:

\[B(\nu,T) = 2kT \nu^2 / c^2 \label{eq17}\]

Stefan-Boltzmann law and luminosity

Integrating over Equ \eqref{eq15} over all directions and frequencies yields the entire energy output of a blackbody per second per square meter, describe by the Stefan-Boltzmann law (\(\sigma = 5.67\)Wm\(^{-2}\)K\(^{-4}\)):

\[E(T) = \sigma T^4 \label{eq18}\]

Note: The total energy emitted by a blackbody depends strongly on temperature.

This gives an expression for the luminosity (the total energy output) of a star, by multiplying Equ \ref{eq18} with the surface area:

\[L = 4 \pi R^2 \sigma T^4 \label{eq19}\]

If two of the three quantities \(L\), \(R\), and \(T\) are known, the third can be derived using Equ \eqref{eq19}.

Colours of stars

Colours for stars are measured using multi-band photometry, i.e. measuring the magnitudes in different regions of the spectrum (’bands’) – see Fig \ref{fig8}. The ’colour’ is the difference between magnitudes in two bands. The colours are related to the temperatures of stars via Wien’s law (Equ \eqref{eq16}).

Hot stars are bright in the blue and faint in the red, i.e. \((B-R)<0\). Cool stars are faint in the blue and bright in the red, i.e. \((B-R)>0\). Colours are defined to be 0.0 for a star like Vega.

Kirchhoff’s laws

Stars emit continouous radiation due to the thermal motion of particles. The continuuum is blackbody radiation, its spectrum is described by the Planck function. But stars are not pure blackbodies. Their spectra show specific regions where less light is emitted than for a blackbody. These regions appear as dark bands in the spectra and are called spectral lines.

According to Kirchhoff’s laws, there are three types of spectra (see Fig \ref{f9}):

  • A continuous spectrum (light at all wavelengths) is produced by hot opaque body, usually approximated by a blackbody.

  • A continuous spectrum with dark absorption lines is produced by a hot opaque body seen through a transparent layer of cool gas which absorbs light at specific wavelengths. This describes a star – the hot blackbody is seen through the cooler atmosphere of the star.

  • An emission line spectrum is produced by hot transparent gas. Here light is only emitted at specific wavelengths. This can occur, for example, when clouds of gas are heated by nearby hot stars.

Origin of spectral lines

Spectral lines are caused by absorption or emission of EMR in atoms or molecules in the atmospheres of stars. An atom can absorb or emit only photons of certain \(\lambda\), when electrons ’jump’ from one energy level to another. The change in energy is proportional to the frequency of the emitted or absorbed photon:

\[E = h \nu = h c / \lambda \label{eq20}\]

The energy differences between the electron levels in atoms are very small, typically in the range of electron Volts (1eV is \(1.602 \cdot 10^{-19}\)J).

Spectral lines in Hydrogen

Hydrogen is the simplest atom, with only one proton and one electron. The energy levels for the Hydrogen electron are shown in Fig \ref{f10}.

The energy difference between ground state (\(n=1\)) and first excited state (\(n=2\)) is \(E_{12} = 10.2\)eV. With Equ \eqref{eq20} this gives \(\lambda_{12} = 121.6\)nm. This line is called Lyman \(\alpha\) and is in the ultraviolet part of the spectrum. Other transitions to or from the ground state have higher energies, i.e. shorter wavelengths. The Lyman limit at \(\lambda = 91.2\)nm corresponds to the energy required to move an electron from the ground state to ’infinity’, i.e. out of the Hydrogen atom.

In general, the wavelength for the photons corresponding to the transitions in the Hydrogen atom are described by the Rydberg equation:

\[1/\lambda = R (1/n_l^2 - 1/n_u^2) \label{eq21}\]

Here, \(R\) is the Rydberg constant (\(R = 1.097 \cdot 10^{-7}\)m\(^{-1}\)). \(n_l\) and \(n_u\) are the quantum numbers of the lower and upper level in a specific transition.

Lyman series: \(n_l = 1\), \(n_u >1\), in the UV Balmer series: \(n_l = 2\), \(n_u >2\)<