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A brief introduction to the origin and interpretation of the Hertzsprung-Russell Diagram is illustrated by reference to some well-known amateur objects. Some more advanced topics – the Hyashi Line, Eddington Limit, Humphreys-Davidson Limit, and Langer's proposed 'Omega' Limit – are briefly surveyed.
Keywords: Hertzsprung-Russell Diagram, Hayashi Line, Eddington Limit, Humphreys-Davidson Limit, rotational instability
Once a system of spectral classification had been conceived, and many thousands of stars classified accordingly by Cannon and her Harvard colleagues, it became possible for the first time to consider their properties en masse. In 1913, Ejnar Hertzsprung (Denmark) and Henry Norris Russell (United States) independently discovered that the absolute luminosity of most stars is very strongly correlated with their temperature.
When colour index or, equivalently, temperature, is plotted against absolute luminosity, stellar properties are displayed in two dimensions and any natural clustering of their properties becomes easy to observe. Such graphs are generally called Hertzsprung-Russell diagrams or HRDs, although the term colour-magnitude diagram is occasionally seen, too. They mean the same thing. A simple example is shown in fig. 1.
The first observation is that stars do not plot randomly all over the diagram, but are mostly restricted to a few well-defined regions. Moreover, were the physical properties of a star to change over time - if the star gradually became cooler (= redder) and dimmer, for example - it's plot on the diagram would move, tracing out a line (although, of course, no physical movement in space is implied by this). But with a few extremely rare exceptions such as supernovas, the properties of stars change so slowly that we do not notice any points moving on the HRD in our lifetime. Even an HRD compiled from observations spanning decades may be safely regarded as representing a point in time.
Yet if the stars we plot there are chosen from all over the sky - not, say, chosen from a single cluster or association - then we can reasonably believe they are of all different ages. If so, then the clustering of plots we see on the HRD must represent configurations in which stars spend a long time, relative to the configurations for which there are very few plots. (If you were to randomly sample all of the aeroplanes in the world at a particular moment, the vast majority would be either on the ground or in the air, with only a small number actually in the act of taking off or landing.)
A Comparison of Some Different Stars
The Zero-Age Main Sequence
On HR diagrams of diverse stars, the majority of plots are found to cluster in a narrow band bordered on its lower edge by the dark, downward-sloping line, known as the zero age main sequence (ZAMS). This is the position on the HR diagram assumed by young stars once they derive most of their energy from thermonuclear reactions, rather than from the release of gravitational energy as the new stars contract from their protostellar gas clouds (Zeilik 1991).
As the axes are conventionally drawn, most "normal" stars plot slightly to the upper right of the ZAMS.
Distribution along the ZAMS is determined by a number of factors, of which mass is by far the most significant: heavier stars, being hotter and more luminous plot further towards the upper left whereas less massive, cooler and dimmer stars plot to the lower right. Another factor, playing a small role, is composition.
On this diagram we can see that the sun is an "ordinary" main sequence star. The other main sequence stars shown are the very massive, hot and luminous Spica (a Vir); and Sirius A (a CMaj A) which, while more massive, is not greatly unlike the sun.
Giants and Supergiants
Although the main sequence accounts for the majority of all stars, the visible sky features many other types as well. One of the most prominent groups is the red giants and supergiants. These are cool yet very luminous stars such as Betelgeuse (a Ori) shown in the figure. Red giants and supergiants have a low surface temperature and only appear bright on account of their massive area.
There is probably no such thing as a "typical" supergiant, but Betelgeuse, for example, has a diameter around 1000 (it fluctuates) times that of our sun (denoted 1000 R¤). This diameter of approximately 3.6 AU, if centred upon our sun, would exceed the orbit of Mars and most of the asteroid belt (most asteroids circle the Sun at distances between 2 and 3.5 AU).
Another group of stars which plot a long way from the ZAMS are the white dwarfs. These are (typically) hot yet also very dim stars – none is visible to the naked eye.
White dwarfs such as Sirius B (a CMaj B) are stars which have mostly consumed their fusionable elements and undergone gravitational collapse. Being insufficiently massive to collapse into a neutron star, further collapse is prevented by electron degeneracy pressure. One consequence of their gravitational collapse is a very small diameter, thus a small surface area and correspondingly low luminosity.
Special Regions on the HR Diagram
The properties and behaviours of stars – especially very large stars – are constrained by exotic physical processes which are only becoming understood today. They manifest themselves on the HR Diagram as a group of limits and zones which one seldom encounters in the ‘amateur’ literature, so a brief survey is given here.
The earlier discussion was necessarily somewhat over-simplified. The "zero-age" in the ZAMS concept implies an evolutionary interpretation which cannot be known from direct observation. Real HR diagrams prepared from stellar observations confirm the clustering of most stars into a relatively narrow band stretching from blue, high-luminosity to red, low-luminosity loci. Stripped of evolutionary interpretation, this band is known simply as the main sequence.
There are alternative possible definitions of 'main sequence' and both have been used in the profesional literature:
Astrophysically, a definition based on stellar structural properties is preferable (Nota et al. 1996, p. 384).
The low luminosity end of the main sequence has a theoretical limit corresponding to a stellar mass of about 0.08 M¤, below which hydrogen burning cannot be sustained in the core, though in practice the lower limit of real HR diagrams is determined by the magnitude limit of the observations.
In 1962, Hayashi, Hoshi & Sugimoto suceeded in giving an approximate analytical relation between the stellar parameters L, M, Teff, E0, k0 and Z (see Böhm-Vitense 1992, Eq. 11.5) for luminosities less than that of the sun (1 L¤). They found that stars of a given mass lie on an essentially vertical line on the HR diagram.
|Proto-stars in the process of contracting towards the ZAMS, prior to the development of a radiative core and the initiation of hydrogen fusion, are fully convective. The actual evolutionary track followed by a contracting star follows the Hayashi line in the beginning, until it develops a growing core in radiative equilibrium when it moves to the left (higher Teff) and finally reaches the ZAMS with the initiation of hydrogen burning.|
|More generally, the coolest stars are always fully convective. Since fully convective stars have lower Teff than those with zones in radiative equilibrium, the Hayashi line gives a lower limit for the Teff of all stars in hydrostatic equilibrium.|
|The region to the right (cool side)
of the Hayashi line is the so-called forbidden zone, where there are no
equilibrium solutions to the equations of stellar structure. The few objects
which are observed to occur in the forbidden zone are assumed to be either:
* Collapsing protostars undergo very rapid time variations, and are examples of objects not in hydrostatic equilibrium. The complete pre-main-sequence evolutionary track from initial molecular cloud with low luminosity and surface temperature to protostar with a luminosity much higher than the main sequence luminosity, does pass through the forbidden zone.
[After Tayler 1994 and Böhm-Vitense 1992.]
Stellar clusters are invaluable in the study of stellar evolution because, approximately, all of their members are at the same distance, allowing direct and immediate observation of their relative brightness, and because they are assumed to be all of the same approximate age. Thus, studies of clusters provide a snapshot of how a group of stars of the same age and with the same initial composition have evolved with time.
|A characteristic feature of the HR diagrams of clusters is the turn-off from the main sequence of stars which have evolved away from the main sequence.||<fig. ?>|
The atoms and ions in a stellar atmosphere are constantly subjected to bombardment by photons, each of which accelerates the particle in the direction the photon was travelling. Although the photons will come from all directions, there will be a net outflow away from the centre of the star and therefore a net outwards acceleration applied to the atmospheric particles. By calculating the values at which these radiative forces become strong enough to destroy the whole star, Eddington determined the maximum possible luminosity a star might possess. Beyond the Eddington Limit, stars would fly apart under their own radiative pressure alone.
|The Eddington parameter, characteristic for each star and denoted G or sometimes Ge, is given by:|
|G = k/4pcG (L/M)||(1)|
|where k is the flux-mean opacity coefficient. The Eddington Limit is where G = 1.|
|For a star like the sun, G is very small, ~ 2 x 10-5, but for very massive stars it may be much higher. "It is certainly significant that hot stars with strong stellar winds have Ge only a factor of two of so below [the Eddington Limit], since it suggests that only a modest additional opacity could succeed in fully overcoming gravity to drive an outflow. But it is important to realise that a stellar wind represents the outer envelope outflow from a nearly static, gravitationally bound base, and as such is not consistent with an entire star exceeding the Eddington limit" (Owocki 2000).|
|For a star of 1 M¤, the maximum possible luminosity is about 5 ´ 104 L¤ or about Mbol = -7.0. For a star of 40 M¤, Mbol » -11.0 and, in fact, we do not observe stars much brighter than this. At Mbol » -11.6, Eta Carinae is one of the most luminous stellar objects of our Galaxy.|
The Eddington Limit is linear yet, as further data for stars near the top of the HR diagram accumulated towards the end of the 1970s (e.g. Hutchings 1976), it became apparent that the upper luminosity limit varies non-linearly with temperature for hot stars. For these early type stars the maximum luminosity decreases with decreasing effective temperature, this downward trend ceasing at around Teff = 104K, after which the maximum luminosity remains more or less constant. Humphreys and Davidson (1979) noted this boundary and suggested that an instability leading to rapid and unsteady mass loss set in there. The observed boundary could, in principle, be explained by steady mass loss, but only if the loss were much higher than that which is actually observed for objects near the boundary (typically 10-5 M¤ per year), so it is the "unsteady" mass loss in sporadic eruptions which is considered to be of foremost importance.
|"A stability limit, probably caused by radiation pressure, is a possibility, but the classical Eddington luminosity limit did not show the observed dependence on temperature for the hot stars. However, as temperatures decrease below about 30,000 K, opacity tends to increase as low-level ions such as HI, FeII, et al., begin to appear; so we can imagine a modified Eddington luminosity limit which decreases with decreasing temperature, like the observed limit" (Humphreys 1989, p. 3).|
|There are still problems with this explanation, however, and the suggestion is still a topic for further research.|
A new idea (Langer 1997) posits that the Eddington Limit never actually applies because a lower limit applies in real (i.e. rotating) stars, which will disintegrate from the centrifugal forces before radiation pressure disperses them. Langer has dubbed this the 'Omega' Limit, defined as:
|W := nrot / ncrit > 1||(2)|
|ncrit2 = GM/R (1 - G)||(3)|
|As far as is currently explored, this argument ignores any potential "gravity darkening" in a rotating star. Analysis dating back to Von Zeipel (1924) suggests that in rotating stars with purely radiative energy transport (i.e. without any convection), the local radiative flux should scale in proportion to the local effective (centrifugally reduced) gravity. In this case, the local Eddington factor (ratio of electron radiative force to effective gravity) is actually constant. Then, contrary to the Omega Limit argument, the equatorial regions would not exceed any lower threshold before the overall stellar Eddington limit.|
|Although it is not certain that Von Zeipel's idealized radiative diffusion analysis applies to such an extreme case, nevertheless, it is the best theory we have to date, so such effects cannot be ignored. Aside from this, there are other questions regarding the assumptions made in this scenario about the rigid body rotation of the expanding stellar envelope (Stan Owocki, pers. comm.)|
As we have noted, mass increases towards the top left of ZAMS and of the HR diagram in general. Where does this end? "There is no accepted upper mass limit for stars. Such a basic quantity eludes both theory and observation, because of an imperfect understanding of the star-formation process and because of incompleteness in surveying the Galaxy" (Figer 2005, p. 192).
Although the Galaxy as a whole remains incompletely surveyed, is has
been possible to exhaustively examine a particular cluster, the Arches
Cluster, which is close enough to observe, large enough to give rise
to supermassive stars, and old enough for the star-forming dust to have
been dispersed, yet young enough for the massive stars to still exist.
Figer's analysis of the data so derived has led him to conclude that his
study "indicates a firm limit of 150 M¤
for stars; the probability that the observations are consistent with there
being no upper limit is
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Chiosi, C.; Maeder, A. 1986: The Evolution of Massive Stars with Mass Loss. Annual Review of Astronomy & Astrophysics 24: 329-75.
Figer, Donald F. 2005: An upper limit to the masses of stars. Nature 434: 192-194.
Humphreys, Roberta M. (1989): What are LBVs? – Their Characteristics and Role in the Upper H-R Diagram. In K. Davidson et al. (eds.), Physics of Luminous Blue Variables, p. 3 - 12.
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