The HR Diagram and Stellar Evolution

The Blackbody Spectra of Stars


Colour_Index
Blackbody spectra
Two Blackbody spectra with B and V bandpasses shown
 

A stellar spectrum consists of a continuum which is roughly shaped like that emitted by a blackbody of some effective temperature, such as those in the diagram above. Remember a blackbody is an object which absorbs all radiation falling upon it and is in thermodynamic equilibrium (i.e. is not heating up or cooling down as a result of the incident radiation). Superposed on this continuum there are absorption lines (and sometimes emission lines).

The Stefan-Boltzmann law tells us the relationship between the luminosity, size and effective temperature of a star: The effective temperature of a star is the temperature of a blackbody the same size as the star which emits the same total power i.e. bolometric luminosity.

Stefan_Boltzmann
 

where R is the radius of the star and σ , the Stefan-Boltzmann constant, is equal to 5.67 x 10-8 W m-2 K-4.

The effective temperature is an estimate of the surface temperature of the star and can be parameterised using the colour index.

Another important relationship for blackbody spectra is Wien's Law, relating the effective temperature of the blackbody to the wavelength at which the emission is strongest,

Wiens_Law
 

Spectral classification

The MK spectral classification of stars is defined according to the relative strengths of various absorption lines and is effectively a sequence of temperatures, see the following table (note `metals' is the term used by astronomers to refer to all elements other than H and He!).

Spectral type Teff (K) Colour Dominant absorption lines
O 20,000+ Hottest blue stars Ionized He, strong UV continuum
B 20-10,000 Hot blue stars Neutral He dominate (He I)
A 10-7,000 Blue/blue-white stars Neutral H (Balmer lines) dominate
F 7-6,000 White stars Singly ionized Ca (CaII), neutral H weaker, other metals
G 6-5,000 Yellow stars Ca II dominates, neutral metals (e.g. FeI)
K 5,000-3,500 Orange-red stars Neutral metals (Ca, Fe) dominate, molecular bands appear
M 3,500-2,000 Coolest red stars Molecular bands dominate (e.g. TiO), neutral metals

[Remember the roman numeral indicates the ionization state e.g. He I is neutral whilst He II is singly ionized. ]

Note that the hotter stars are sometimes termed 'early-type', the cooler 'late-type'. Each spectral type is further sub-divided into 10 sub-classes denoted 0-9.

A second parameter defining the observed properties of stars is the luminosity class also defined by the ratio of strengths of various pairs of absorption lines. In effect this is a sequence of density in the stellar atmosphere - stars with high luminosity for the same effective temperature are larger and therefore have more tenuous, less dense, outer layers. They follow the scheme given below.

Ia Ib II III IV V VI VII
Bright Faint
low density high density
Bright supergiants Supergiants Bright giants Giants Subgiants Main sequence Sub-dwarfs White dwarfs

The Sun is of type G2V with effective temperature 5800K.

Summary:

Comparing properties of stars

The sun is a fairly typical star. The basic properties of the sun and a comparison with other stars are given in the following table (note the subscript O indicates a property of the sun).

The Sun Range for other stars
LO = 4 x 1026 W 10-4 < L / LO < 106
TeffO = 5800 K 1/3 < Teff / TeffO < 20
RO = 7 x 108 m 10-2 < R / RO < 103
MO = 2 x 1030 kg 10-1 < M / MO < 102

As you can see, Luminosity has the most extreme range and effective temperature the least extreme.

However, stars are not distributed evenly within these ranges. There are in fact two major relationships: luminosity with effective temperature and luminosity with mass. The first of these is usually shown in a Hertzprung-Russell (or HR) diagram.

The Hertzprung-Russell Diagram

In 1911, Ejnar Hertzprung plotted the first diagram of the relative magnitudes of stars in a cluster versus their spectral types. Two years later Henry Russell, working independently, produced a plot of the absolute magnitude of nearby stars with well-determined distances against their spectral types. The Hertzprung-Russell or HR diagram is one of the most useful diagrams in astronomy and is fundamental to our understanding of stars.

The HR diagram is plotted with intrinsic brightness (luminosity or absolute magnitude) on the vertical axis (ordinate), increasing upwards, and effective temperature (colour or spectral type) on the horizontal axis (abcissa), increasing to the left. The most common HR diagrams are called colour-magnitude diagrams after the quantities which are plotted. Look at the simplified example below.

HRDiagram
A Schematic H-R Diagram
 

Understanding the HR Diagram

The main points to note are that 90% of stars lie on a narrow diagonal band running from top left (bright and hot) to bottom right (faint and cool). This is called the Main Sequence (MS). The Sun lies approximately in the middle of the main sequence.

To the right of the MS are stars which are cool (red) but luminous. Since luminosity only depends upon temperature and size, these must be much larger than MS stars of the same temperature. These are the Giants and Supergiants.

In the opposite corner are faint, blue stars - ie hot, small stars or White Dwarfs

  Mass_Luminosity
 

The narrowness of the main sequence reflects the fact that for stars on the main sequence there is a tight correlation between luminosity and mass - the so-called mass-luminosity relation. The image here shows this relation for main sequence stars. The luminosities and masses are plotted on a logarithmic scale relative to solar values. The masses have been determined by analysing the motions of stars in binary systems - this is the most reliable method of determining the mass of a star and is used, for example, to try and find black holes.

Determining Stellar Masses in Binary Systems

Consider two stars of mass M1 and M2 moving in circles of radius a1 and a2 around a common centre of mass, then M1a1 = M2a2. If one can measure a1 and a2 then one can determine the ratio of masses without knowing the distance to the system. Using Kepler's 3rd law, then a measurement of the period P and the separation of the two stars a, leads to knowledge of the sum of the masses and hence the individual masses. However it is not always possible to do this fully.

Double stars are called visual binaries if they can be resolved with a telescope. Sirius is the brightest member of a visual binary with a period of 50 years. Its companion, Sirius B, is a white dwarf. In these cases it is possible to determine the individual stellar masses. Very good orbits have been determined for about 25 visual binaries - these are the basis of the mass-luminosity relation presented above. Some 600-700 orbits are known to some degree of accuracy out of about 50,000 known systems.

Many visually unresolved binaries may have their binary nature revealed if they are sufficiently close together that their orbital speeds exceed about 1 km s-1 so that doppler shifts of spectral lines can be distinguished, these are called spectroscopic binaries. As the inclination of the system cannot be determined it is not possible to determine the individual stellar masses.

In eclipsing binaries one star passes in front of the other and therefore the inclination is known to be about 90° .

In these systems we can determine the masses and radii (from observation of the points at which the light curve dips) of each component, the luminosities (if Teff is known) and hence the distance. They are therefore very useful. However, although 4000 eclipsing systems are known, only about 400 have had their orbits determined and only a small fraction of these to a degree of accuracy which allows determination of all their properties.


Therefore, position on the Main Sequence depends almost entirely upon mass - more massive stars are hotter and more luminous and so at the upper end of the MS.

Stellar Evolution

HR Diagrams can be used to illustrate many of the stages in the life of a star - Stellar Evolution.

Since 90% of stars are on the Main Sequence (MS), it follows that they must spend about 90% of their life there. The rest of the HR diagram is populated by stars before, or more often after their MS phase.

It is important to remember that the Main Sequence is not an evolutionary sequence. The position of a star depends almost entirely on its mass - it will move very little during its MS lifetime.

The formation of main-sequence stars

Triggering collapse

Molecular clouds in the interstellar medium must maintain a balance between internal pressure due to heat or turbulent motions and their own gravity. If this is disturbed (perhaps by a supernova shock wave, see later) then the large cloud begins to fragment into collapsing sub-clouds.

Gravitational collapse

A dense core is formed with material raining in from the surrounding cloud. At this point the protostar is still surrounded by gas and dust and is visible only as an infrared source.

T-Tauri phase

When the remnant cloud around the protostar has thinned sufficiently it may be seen, surrounded by a cloud of gas and dust, this is called the T Tauri phase. It may be associated with an intense wind.

Nuclear ignition

The protostar continues to heat gradually as the contraction continues until the core is hot enough to initiate nuclear reactions. Contraction stops because this energy source can balance gravity. The star is now on the main sequence.

The cloud is initially far to the right and below a normal HR diagram. As the protostar is formed it enters the HR diagram from the right and falls onto the main sequence.

Life on the main-sequence

Once the star is on the main sequence, it will remain there until all the hydrogen in its core has been burnt to helium. This is 90% of the star's lifetime.

We can consider a "zero-age main-sequence" star (or ZAMS star) as a spherical, chemically homogeneous ball of gas in thermal and hydrostatic equilibrium, with nuclear reactions just beginning to alter the chemical composition in the core. Hydrostatic equilibrium is the balance between gravity and internal pressure.

Energy is transported from the core to the surface by one of three methods: conduction, radiation and convection. Conduction and radiation are similar physical processes, except that in the former particles like atoms, ions, protons, electrons carry energy via collisions whereas in the latter it is photons that transfer energy. However, conduction is almost entirely negligible as the mean free path between encounters is 10-10 m whereas that of photons is 10-2 m. The third form of energy transport is convection; this is not a fully understood process as it is turbulent and remains a major research area.

The energy is generated by nuclear fusion of hydrogen into helium, where the rate of energy generated per unit mass is proportional to T4 (proton-proton chain) for stars like the Sun or of lower mass and proportional to T15 (Carbon-Nitrogen-Oxygen or CNO cycle) in more massive stars.

Evolution off the main sequence

90% of a star's lifetime is spent burning hydrogen on the main sequence. During this period the core gradually gets converted from H to He and contracts slowly, gets hotter and the star increases in luminosity. Thus the star moves upwards slightly on the HR diagram and thus broadens the main sequence. The last 10% of the stellar lifetime is much more eventful.

The evolution of a star is a constant battle to avoid collapse under its own gravity. It does this by generating internal energy via nuclear reactions. First by burning hydrogen, then helium. Whether further elements are available as fuel depends on the mass of the star. Stars of mass greater than about 8 MO can go on to burn carbon and heavier elements.

One important consequence of this battle is that massive stars have much shorter MS lives than less massive stars. Although they have more hydrogen "fuel" to start with, they burn it much more quickly. So, where the Sun has an MS lifetime of about 1010 years, a 25 MO star will have an MS lifetime of a few times 106 years and a 0.5 MO star will be on the Main Sequence for 2 x 1011 years.

After the Main Sequence, the star goes through several stages of evolution which are briefly described here for a star like our Sun.

Stellar_Evolution_Sun_like
Evolution of a Sun-like star
 

Depletion of hydrogen in the core and the onset of hydrogen shell burning

Eventually all the hydrogen in the core where the temperatures are high enough to support nuclear reactions is depleted. The core, in which there is now no energy generation, slowly contracts bringing fresh hydrogen closer to the centre where temperatures increase to the point that nuclear reactions begin in a shell around the inert helium core. The radiation produced by the shell source causes the outer layers of the star to expand outwards causing the star to move upwards and to the right on the HR diagram (ascending the giant branch).

Development of a degenerate core

As the core contracts, the density increases to the point where quantum effects become important - the Pauli exclusion principle prevents packing electrons extremely close together and there is therefore an extra contribution of pressure referred to as degeneracy pressure when the gas is said to be degenerate. This pressure is independent of temperature and depends only on the density, unlike in an ideal gas. In stars with masses less than 2 MO the helium core becomes degenerate before the temperature is high enough to initiate helium burning.

The helium flash and the horizontal branch

When the temperature of the core reaches 108 K, three helium nuclei can begin to fuse to form carbon - the triple alpha reaction. This generates energy at a rate proportional to T40! In addition, carbon nuclei can combine with helium nuclei to form oxygen. In stars of mass less than 2 MO, helium burning rapidly increases the temperature of the core. Because the core is degenerate, there is no safety valve in which this temperature increase results in an increased pressure which would cause the core to expand, cool, and reduce the rate of energy production. In fact, there is a runaway in which helium burning occurs explosively - the 'helium flash'. Eventually when the temperature is so high that the ion pressure is high enough to expand the core, degeneracy is lifted and helium burning continues in a more stable fashion. The star moves to the left (gets hotter) once helium burning starts and moves to a point on the horizontal branch which constitutes the main sequence for helium burning. The star now consists of a helium-burning core with a hydrogen burning shell.

Up the asymptotic giant branch

Once the helium is exhausted in the fully convective core, the whole star contracts. The star now consists of a carbon-oxygen core and a helium shell surrounded by an envelope of mostly hydrogen. The core continues to contract and becomes degenerate but for stars of mass less than 8 MO it never becomes hot enough to initiate nuclear reactions. Nuclear burning starts again when the base of the helium shell is brought in to high enough temperatures. The star now moves back up the HR diagram and asymptotically approaches the giant branch - it is therefore referred to as an Asymptotic Giant Branch (or AGB) star. There can be two shell sources, one helium and one hydrogen. Gradually the AGB stars evolve towards the tip of the AGB.

Planetary nebulae and white dwarfs

At the tip of the AGB the star is very extended and the surface layers are only loosely bound to the star. The hydrogen envelope is ejected at a speed of a few tens of kilometres per second. The remnant star consists of a carbon-oxygen core surrounded by a shell of unburnt helium. It is still a supergiant but has moved across the HR diagram to the left, becoming blue rather than red. This hot star ionizes the ejected shell causing it to glow in optical and ultraviolet emission lines - it is then called a planetary nebula (because in small telescopes they look a bit like fuzzy planets). The central star contracts until degeneracy pressure brings it into equilibrium. It is now a white dwarf that spends the rest of its life gradually cooling.

Each of the stages described above is shown on the HR Diagram at the top of this section.

HR diagrams for clusters

The problem with HR diagrams is that for them to make sense we need to know the absolute magnitudes or intrinsic luminosities of the stars. This means we need to know their distances - often not a straightforward task. We can get round this problem by looking at the HR diagrams of clusters of stars i.e. groups of stars which are all at approximately the same distance. In these cases we can just use their apparent magnitudes in HR diagrams as distance will have affected all stars in the same way and simply caused them to shift vertically in the diagram.

There are two types of star cluster:

Open (or galactic) clusters

Loose structures containing about 100 to 1000 stars of similar chemical composition to the Sun, the brightest of which are mostly blue. Irregularly shaped, ranging in size from 1-20 parsecs (pc). Almost a thousand are known in our Galaxy and are mainly confined to the disc e.g. Pleiades, Praesepe. Some contain gas and dust, others don't. Shown at left below is a negative image of the open cluster Praesepe sometimes known as the Beehive (a 1x1 degree field obtained from the Digitized Sky Survey)

Globular clusters

Very different to open clusters. Centrally condensed, virtually spherical, containing about 100,000 to 1,000,000 stars, the brightest of which are red. Typically about 40 pc across. The 150 known in our Galaxy are distributed roughly spherically about the centre of the galaxy. They do not appear to contain gas or dust. The stars are typically underabundant in heavy elements by a factor of 10-1000 in comparison to the Sun. Shown at right below is an image of the globular cluster M3 (a 30x30 arcminute field from the Digitized Sky Survey).

Open_and_Globular_Clusters
Example Open (left) and Globular (right) Clusters
 

The assumption is that all stars in a cluster were formed at the same time out of the same material - they are therefore all the same age and have the same chemical composition. The only thing that should distinguish them is their mass.

In fact, the HR diagrams for open clusters and globular clusters look very different. For the open cluster, most stars lie on the main sequence with a few stars in the giant region above the main sequence and perhaps a few stars in the white dwarf region, see example below for Praesepe. However, the actual number of stars in the main sequence and in the giant region does vary from cluster to cluster.

Praesepe_HR_Diagram
 

For the globular cluster, the main sequence is much less apparent. There is a well-populated giant branch and horizontal branch. Most globular cluster HR diagrams look very similar, see the example below for M3 (ignore the "Isochrone" line).

M3_HR_Diagram
 

These differences are interpreted as age differences. Globular clusters are thought to all be within a factor of two of 10 billion years old, while open clusters are between a million and 10 billion years old.

In the youngest clusters virtually all stars are on the main sequence. The length of time a star spends on the main sequence, Tms, depends on its mass and the rate at which it burns its fuel i.e. its luminosity. In fact, Tms = 10 billion years x (M / L) where 10 billion years is the theoretical main-sequence lifetime for the Sun and M and L are, respectively, the mass and luminosity of the star in units of the Sun's mass and luminosity. We know that the mass-luminosity relation for main sequence stars tells us that their luminosity is proportional to almost the fourth power of their mass, therefore more massive stars spend less time on the main sequence before becoming giants.

As a cluster ages, the more massive bluer stars peel off the main sequence first and head to the right into the giant region. Later, the less massive redder stars evolve into the giant region. At a given moment when we observe a cluster the stars at the top of the main sequence (the turn-off point) are those which have just reached the end of their main-sequence lifetimes. Therefore, if we measure the luminosity of the turn-off point on the main sequence, and use the mass-luminosity relation to determine the mass, then the above equation tells us the age of the cluster.

Example - A galactic cluster has a well-defined main sequence with a turn-off point at L = 81 LO. Assuming that the mass-luminosity relation for stars near the turn-off point is L/LO = (M/MO)4, and that the main-sequence lifetime of the Sun is 1010 years, estimate the age of the cluster. (ANSWER 3.7 x 108 years).

An alternative way of getting ages is to compare the observed HR diagram with a theoretical one calculated using mathematical models of the structure and evolution of stars. The line labelled "10 Gyr isochrone" on the M3 HR diagram represents the results of such a calculation. An isochrone is simply a locus connecting model stars of various masses but at the same age. One can vary various parameters in such models in addition to the age such as chemical composition, the effects of stellar mass-loss etc.

Normally such models would output absolute visual magnitude rather than apparent magnitudes. The example isochrone shown has been shifted vertically relative to the observed HR diagram. What does the amount of shift tell us about the cluster?

The models will also not have included interstellar reddening - what effect would this have on the comparison between model and observation?