Stellar radiation and stellar types-State that fusion is the main energy source of stars-Explain that, in a stable star, there is an equilibrium between radiation pressure and gravitational pressure.-Define the luminosity of a star.-Define apparent brightness and state how it is measured.-Apply the Stefan-Boltzmann law to compare the luminosity of different stars.-State Wien’s (displacement) law and apply it to explain the connection between the colour and temperature of stars.-Explain how atomic spectra may be used to deduce chemical and physical data for stars.-Describe the overall classification system of spectral classes.-Describe the different types of star.-Discuss the characteristics of spectroscopic and eclipsing binary stars.-Identify the general regions of star types on a Hertzsprung-Russell diagram

Nuclear fusion in the starsA star such as our own Sun radiates an enormous amount of energy into space—about 1026 J/s. The source of this energy is nuclear fusion in the interior of the star, in which nuclei of hydrogen fuse to produce helium and release energy in the process.

Because of the high temperatures in the interior of the star, the electrostatic repulsion between protons can be overcome and hydrogen nuclei can fuse.

Because of the high pressure in stellar interiors, the nuclei are sufficiently close to each other to give a high probability of collision and hence fusion.

FusionThe sequence of nuclear fusion reactions that take place is called the proton-proton cycle, and consists of:

Note that the net effect of these reactions is to turn 4 hidrogen nuclei into one helium:

The energy released per reaction is around 4 x 10-12 J.

Radiation Pressure

The energy produced is carried away by the photons and neutrinos produced in the reactions. As these particles move outwards they collide with surrounding protons and electrons and give them some of the energy. Thus, gradually, most of the particles in the star will receive some of the kinetic energy produced. The motion of the particles inside the star, as a result of the energy they receive, can stabilize the star against gravitational collapse.

Luminosity

Luminosity (L) is the amount of energy radiated by the star per unit of time; that is, it is the power radiated by the star. Its unit is W=J/s.

Luminosity depends on the surface temperature and suface area of the star.

Apparent brightnessConsider a star of luminosity L. Imagine a sphere of radius d centered at the location of the star. If the star is assumed to radiate uniformly in all directios, then the energy radiated in a unit of time can be thought to be distributed over the surface of this imaginary sphere. A detector of area a placed somewhere on this sphere will receive a small fraction of this total energy. The fraction is equal to the ratio of the detector area a to the total surface area of the sphere; that is the received energy per unit of time will be aL/4πd2.

The received energy per unit of time per unit of area is called the apparent brightness and is given by

b = L/4πd2.The units of apparent brightness are W/m2

Black-body radiationBlack-body is any object that is a perfect emitter and a perfect absorber of radiation

A black-body of surface area A and absolute temperature T radiates energy away in the form of electromagnetic waves at a rate given by the Stefan-Boltzmann law

L = σAT4,where σ is called the Stefan-

Boltzmann constant

σ = 5.67 x 10-8 W/m2K4

Thus,

b = σAT4/4πd2

Wien law

Most of the energy is emitted around the peak wavelength. Calling this wavelength λ0, we see that the colour of the star is mainly determined by the colour corresponding to λ0.

The Wien displacement law relates the wavelength λ0 to the surface temperature T:

λ0T = constant = 2.9 x 10-3 Km

Example questions1. The radius of star A is three times that of star B and its temperature is double that of B. Find the ratio of the luminosity of A to that of B.

2. The stars in question 1 have the same apparent brightness when viewed form earth. Calculate the ratio of their distances.

3. A star has half the Sun’s surface temperature and 400 times its luminosity. How many times bigger is it?

4. The Sun has an approximate black-body spectrum with most of the energy radiated at a wavelength of 5 x 10-7 m. Find the surface temperature of the Sun.

Radiation from starsThe crucial point for astronomers was that stars emit light waves, and they hoped that the wavelengths could tell them something about the stars.

For example, once an object reaches 500°C it has just enough energy to emit visible red light, and is literally red hot. As the temperature increases, the object has more energy and emits a greater proportion of higher-energy, shorter, bluer wavelengths and it transforms from red hot to white hot, because it is now emitting a variety of wavelength from red to blue. The filament of a standard light bulb operates at approximately 3,000°C, which certainly makes it white hot.

SpectroscopyAs well as measuring the temperature of a star, astronomers worked out how to analyse starlight in order to identify a star’s ingredients.

Thomas Melvill subjected various substances to a flame and noticed that each one produced a characteristic colour. For example, table salt gave off a bright orange flash of colour. The distinctive colour associated with salt can be traced to its structure at the atomic level. The orange light is generated by the sodium atoms. By passing the light from sodium through a prism, it is possible to analyse exactly which wavelengths are emitted, and the two dominant emissions are both in the orange region of the spectrum.

The exact wavelengths emitted by each atom act as a fingerprint. So by studying the wavelengths emitted by a heated substance, it is possible to identify the atoms in the substance.

SpectroscopyThe process by which a substance emits light is called spectroscopic emission. The opposite process, spectroscopic absortion, also exists, and this is when specific wavelengths of light are absorbed by an atom. So, if a whole range of wavelengths of light were directed at vaporised salt, then most of the light would pass through unaffected, but a few key wavelengths would be absorbed by the sodium atoms in the salt. The absorbed wavelengths for sodium are exactly the same as the emitted wavelengths, and this symmetry between absorption and emission is true for all atoms.The Sun is hot enough to emit wavelengths over the entire range of visible light, but physicists at the start of the 19th century noticed that specific wavelengths were missing. These wavelengths revealed themselves as fine black lines in the solar spectrum. The missing wavelengths had been absorbed by atoms in the Sun’s atmosphere. Indeed, the missing wavelengths could be used to identify the atoms that make up the Sun’s atmosphere.

SpectroscopyThis technique of stellar spectroscopy was so powerful that in 1868 British Norman Lockyer and French Jules Janssen independently discovered an element in the Sun before it was discovered in Earth. The identified an absortion line in sun-light that could not be matched with any known atom. It was named helium, after Helios, the Greek sun-god. Although helium accounts for a quarter of the Sun’s mass, it is very rare on Earth and it would be over twenty-five years before it was detected here.

It has been found, however, that most stars have essentially the same chemical composition, yet show different absorption spectra. The reason for this difference is that different stars have different temperatures. Consider two stars with the same content of hydrogen. One is hot, about 25000 K, and the other cool, about 10000 K. The hydrogen in the hot star is ionized , which means the electrons have left the hydrogen atoms. These atoms cannot absorb the photons and make transitions to higher energy states. Thus, the hot star will not show any absorption lines at hydrogen wavelengths.

Proper motionFollowing Galileo, astronomers had assumed that the stars were stationary. Although the stars all moved across the sky every night, astronomers realised that this apparent motion was caused by the Earth’s rotation.

Edmund Halley became aware of subtle discrepancies in the recorded positions of the stars Sirius, Arcturus and Procyon compared with measurements made by Ptolomy many centuries earlier. Halley realised that these differences were not down to inaccurate measurements, but were the result of genuine shifts in the positions of these stars over time.

In general, detecting proper motion has required careful observations of the closest stars taken across several years.

Radial velocity and Doppler effectProper motion is a measure of motion across the sky only, and says nothing about motion towards or away from the Earth, known as radial velocity. In order to get this information we need to combine spectroscopy with the Doppler effect.

In 1842 Doppler announced that the movement of an object would affect any waves it was emitting. When an object emitting waves moves towards an observer, then the observer perceives a decrease in the wavelength, whereas when the emitter moves away form the observer, then the observer perceives an increase in the wavelength. Alternatively, the emitter might be stationary and the observer might be moving, in which case the same effects are apparent.

Radial velocity and Doppler effectThe three spectra show how the light emitted by a star depends on its radial motion. The spectrum in the middle shows the wavelength of some absorption lines from a star which is neither moving closer to nor farther from the Earth. The spectrum below shows redshifted absorption lines from a star which is moving away from Earth-the lines are identical, except they have all been shifted to the right. The spectrum above shows blueshifted absorption lines from a star which is moving towards the Earth-again, the lines are identical, except this time they have all been shifted to the left. Measurement of the shift allows the determination of the radial velocity of the star.

Spectral Class

Spectral Class Colour Temperature/K

O Blue 25 000 – 50 000

B Blue - white 12 000 – 25 000

A White 7 500 – 12 000

F Yellow - white 6 000 – 7 500

G Yellow 4 500 – 6 000

K Yellow - red 3 000 – 4 500

M Red 2 000 – 3 000

The Hertzsprung-Russell diagramAstronomers realized that there was a correlation between the luminosity of a star and its surface temperature. The Danish astronomer, Ejnar Hertzsprung plotted luminosities versus surface temperature and the American, Henry Russell, plotted absolute magnitude versus spectral class. Such plots are now called HR diagrams.

In the HR diagram, the vertical axis represents luminosity in units of the sun’s luminosity (i.e., 1 corresponds to the solar luminosity of 3.9 x 1026 W). The horizontal axis shows the surface temperature of the star. Also shown at the top of the diagram is the spectral class for each star, which is an alternative way to label the horizontal axis. The scale on the axes is not linear.

HR diagramThree clear features emerge from the HR diagram:

1. Most stars fall on a strip extending diagonally across the diagram from top left to bottom right. This is called the main sequence.

2. Some large stars, reddish in colour, occupy the top right – these are the red giants (large, cool stars)

3. The bottom left is a region of small stars known as white dwarfs (small and hot).

In fact, about 90% of all stars are main sequence stars, 9% are white dwarfs and 1% are red giants. Another feature of the HR diagram is that, as we move along the main sequence toward hotter stars, the mass of the star increases as well. Thus, the right end of the main sequence is occupied by red dwarfs and the left by blue giants.

Main sequence stars

Main sequence stars produce enough energy in their core, from nuclear fusion of hydrogen into helium, to exactly counterbalance the tendency of the star to collapse under its own weight.

Our Sun is a typical member of the main sequence.

Red giantsRed giants are very large, cool stars with a reddish appearance. The luminosity of red giants is considerably greater than the luminosity of main sequence stars of the same temperature; they can, in fact, be a million or even a billion times bigger. The mass of a red giant can be as much as 1000 times the mass of the Sun, but their huge size also implies small densities. In fact, a red giant will have a central hot core surrounded by an enormous envelope of extremely teneous gas.

White dwarfsWhite dwarfs form when a star collapsing under its own gravitation stabilizes as a result of electron degeneracy pressure. This means that the electrons of the star are forced into the same quantum states. To avoid that, the Pauli exclusion principle forces them to acquire large kinetic energies. The large electrons energies can then withstand the gravitational pressure of the star.

An example is Sirius B, a star of mass roughly that of the Sun with a size similar to that of the Earth. This means that its density is about 106 times the density of the Earth.

Stellar evolution

http://highered.mcgraw-hill.com/olcweb/cgi/pluginpop.cgi?it=swf::800::600::/sites/dl/free/0072482621/78780/HR_Nav.swf::H-R%20Diagram

Variable stars

A variable star is a star whose brightness as seen from Earth fluctuates.This variation may be caused by a change in emitted light or by something partly blocking the light, so variable stars are classified as either:Intrinsic variables, whose luminosity actually changes; for example, because the star periodically swells and shrinks.Extrinsic variables, whose apparent changes in brightness are due to changes in the amount of their light that can reach Earth; for example, because the star has an orbiting companion that sometimes eclipses it.

Cepheid starsThis group consists of several kinds of pulsating stars that swell and shrink very regularly by the star's own mass resonance. Generally the Eddington valve mechanism for pulsating variables is believed to account for cepheid-like pulsations: a certain helium layer of the star has variable opacity depending on the ionization degree, greater opacity for the greater level of ionization. At minimum the star is contracted so that the layer has the higher ionization and opacity, and therefore absorbs fusion energy for the star to expand. When the star swells up to a certain size, the ionization suddenly switches from higher to lower, switching the opacity to lower too. The inner fusion energy now radiates more easily through this star layer, so the star shrinks to the original contracted state, and the cycle begins anew.

Cepheids are important because they are a type of standard candle. Their luminosity is directly related to their period of variation. The longer the pulsation period, the more luminous the star. Once this period-luminosity relationship is calibrated, the luminosity of a given Cepheid whose period is known can be established. Their distance is then easily found from their apparent brightness. Observations of Cepheid variables are very important for determining distances to galaxies within the Local Group and beyond. A relationship between the period and luminosity for classical Cepheids was discovered in 1908 by Henrietta Swan Leavitt in an investigation of thousands of variable stars.

Binary stars

A system of two stars that orbits a common centre is called a binary sistem. Depending on the method used to observe them, binaries fall into three classes:

• visual• eclipsing• spectroscopic

Visual binaries

These appear as two separate stars when viewed through a telescope. They are in orbit, around a common centre, the centre of mass of the two stars.

The common period of rotation for a binary is given by

Where d is the distance between the two stars.

The inner star is the most massive of the two.

The two stars are always diametrically opposite each other.

Eclipsing binariesIf the plane of the two stars is suitably oriented relative to that of the Earth, the light from one of the stars in the binary may be blocked by the other, resulting in an eclipse of the star, which may be total or partial. If a bright star is orbited by a dimmer companion, the light curve has a similar pattern to the one shown in the figure.

Spectroscopic binariesThis system is detected by analysing the light from one or both of its members and observing that there is a periodic Doppler shifting of the lines in the spectrum. A blueshift is expected as the star approaches the Earth and a redshift as it moves away from the Earth in its orbit around its companion.

If λ0 is the wavelength of a spectral line and λ the wavelength received on Earth, the shift, z, of the star is defined as

z = | λ – λ0 |/ λ0.

If the speed of the source is small compared with the speed of light it can be shown that

z = v/c