Abstract
This is the second part of a three part series on the subject of measuring stellar parameters by observation. Using data obtained by WMA members and data obtained from the European Space Agency GAIA mission, I revisit the Hertzsprung-Russel diagram (hereafter "H-R diagram") in some detail, showing its use as a tool to classify stages of stellar evolution. I find that, although the scientific principles underlying the H-R diagram were discovered over 100 years ago, and the H-R diagram itself was first used in a paper published in 1911, it is still of fundamental importance today.

Introduction
The H-R diagram is one of the most significant diagrams in astronomy. It was independently discovered by Danish astronomer Ejnar Hertzsprung (1873-1967) and US astronomer Henry Norris Russell (1877-1957) in the early 20th century. Both Hertzsprung and Russell continued prominent astrophysics careers in later years.
In 1909, Karl Schwarzschild (1873-1916), the Director of Potsdam Observatory, offered Hertzsprung the position of Senior Astronomer. Hertzsprung established the concept of absolute magnitude as a method of calibrating astronomical distances. Absolute magnitude is defined as the brightness of a star at the distance of 10 parsecs (32.6 light years). Absolute magnitude can be specified for different wavelength ranges. For stars, the absolute visual magnitude is usually expressed as that in the visual (V) band of the spectrum in the Johnson photometric system.

At Potsdam, Hertzsprung was able to establish a formal relationship between a stars observed colour, which is determined by the photospheric temperature, and its observed absolute magnitude, a measure of the stars luminosity. This laid the scientific foundation for what was to become the H-R diagram. Hertzsprung's first paper on the subject was published in 1911.
Meanwhile, working independently in the United States, Henry Norris Russell (see References) had been studying the same field as Hertzsprung, and came up with very similar conclusions (Russell, Henry N)  . Russell's early version of the H-R diagram, published in 1913, included giant stars, nearby stars with distances measured using the parallax method, and stars in the Hyades open cluster.


Forms of the H-R diagram

1. Colour-magnitude H–R diagram
There are several forms of the H-R diagram. The form of the H-R diagram shown in Figure 2 below is called a colour-magnitude diagram, in this case that of M45, the Pleiades.

We can see from this diagram that the stars lie in a diagonal distribution in absolute magnitude vs. colour index.
The absolute magnitude is magnitude the star will have when viewed from a distance of 10 parsecs. Table 1 below shows the Johnson photometric classification system, the usual form of colour index, derived from various filters.

The colour index is a simple numerical expression that determines the colour of a star which directly scales the stars photospheric temperature. The lower the colour index, the more blue the star is (higher temperature); the higher the colour index, the more red the star is (lower temperature). This form of the H-R diagram is also called the observational H-R diagram.

2. Theoretical H–R diagram
Another form of the H-R diagram is shown in Figure 3 below, in this case of globular cluster M2.

This form of the diagram plots the photospheric temperature of the star on the x-axis and the luminosity of the star on the y-axis, normally as a log-log plot. This form of the H-R diagram is called the theoretical Hertzsprung–Russell diagram.
Data to construct H-R diagrams is freely available online, for example from the European Space Agency GAIA satellite, where photospheric temperature, and luminosity expressed in units of Solar luminosity, have been calculated before being added to the online database

3. H-R diagrams of star clusters
Open Clusters and Globular Clusters are distinctly different types of object. The contrasting H-R diagrams of the Pleiades open cluster M45 and the globular cluster M14 for comparison in Figure 4 below.

We can see that in the case of M45 most of the stars lie in a diagonal distribution from top left to right bottom of the plot. These stars are very young and are said to be on the main sequence, where nuclear reactions within the stellar core are fusing Hydrogen atoms into Helium atoms. This happens due to the intense pressure within the stellar core raising the core temperature to the point that fusion can commence. Note that nuclear fusion happens because the core is hot, and not the other way round. The star’s position along the main sequence is determined by its mass, with the most massive being at the top left, and the least massive at the bottom right.  Note also the length of time a star stays on the main sequence also depends on its mass. The higher the stellar mass, the shorter time the star stays on the main sequence.  Stars at the top left in an H-R diagram spend the least time on the main sequence and stars at the lower right of an H-R diagram the longest time.
It's important to understand that as they evolve, stars do not move along the main sequence; they move off the main sequence once the hydrogen in the star's core has been fused. After the hydrogen in the core of the star has been fused into helium, the star evolves off the main sequence, as we see in the H-R diagram for globular cluster M14 in Figure 4.
Generally, in all H-R diagrams, there is the absence of stars in the region between spectral class A5 and G0 and between absolute magnitude 1 and 3.  This is called Hertzsprung gap.  This is because stars in this region move rapidly through this section – just a few thousands of years.
In the M45 plot in Figure 4, there an obvious outlier - a data point at the bottom right. This is likely to be a white dwarf – the remains of the of an approximately solar mass star at the end of its life.  This star too has come off the main sequence.  Most likely this star is not part of M45 but just happens to be on a similar line of sight.
Globular clusters consist of very old stars - in fact globular clusters are thought to be among the oldest structures in any galaxy. In the H-R diagram of globular cluster M14, few main sequence stars are evident; most stars have left the main sequence having completed hydrogen burning.
Low mass stars in M14 are either ascending the red giant branch or have already become red giants. Like Altair, Vega, and the Sun, they will end their lives as white dwarfs. A few at the top right of the H-R diagram are supergiants and, like Betelgeuse and Deneb, will finish their lives in Type II supernova events.
Globular clusters are found in the halo and nuclear bulge of galaxies. Our galaxy, the Milky Way has 147 identified globular clusters, which is thought to be around half the number that exist. Today, globular clusters have been observed in many other galaxies as well as the Milky Way.
Compared to our region in the disk of the MW, globular clusters are very compact structures. We may deduce this from the density of data points in H-R diagrams for globular clusters, bearing in mind that all he plots in this paper use the same selection parameters (cone angle and magnitude). The number density of stars in the region of the Solar system is only about 0.004 stars per cubic light year. In the dense center of a GC, the number density of stars can be between 500 to 1000 times greater. This compactness results in a very close gravitational binding between stars, which in turn means GCs are very stable. Hence It thought that most GCs will probably maintain their identity almost indefinitely.


Analysing globular cluster H-R diagrams
In Figure 5 below are three globular clusters imaged by WMA members.

We may deduce from Figure 5 that M2 consists of the oldest stars in this sample (i.e. the most highly evolved), M92 the youngest, and M10 somewhere in between. Further examples of globular clusters with stars at various stages of post- main sequence development are shown in Figure 6 below.

Examining the H-R diagrams in Figure 6 above, we can see three globular clusters where most of the constituent stars are well evolved away from the main sequence. 2Mass-GC01 is an extremely faint Milky Way GC. Effectively obscured in visible light by an estimated 21.5 +/- 1.0 magnitudes, it could only be discovered in the infrared.
M12 is approximately 5kpc distant, and has a diameter of about 75 light-years. Comparing this to the diameter of the Milky Way (~100,000 lyr) puts the size of GCs into context, Laevens 1 is the most distant Milky Way globular cluster yet known, at an estimated 146kpc.
In contrast, objects such as M92 and Willman 1 have less stars in post main sequence stages. This can lead to ambiguity in classification. For example, Willman 1 is either a globular cluster or by far the smallest galaxy yet identified (Willman et al). Either way, it is external to the Milky Way and approximately 38 +/- 7 kpc distant.  On the other hand, M92 is confirmed as a Milky Way globular cluster 8.2 kpc distant.

Conclusions
H-R diagrams have been available as astronomical tools for over 100 years and remain of key scientific importance today.  Data for constructing H-R diagrams are freely available online, for example from the European Space Agency GAIA mission (see Acknowledgements).
Using H-R diagrams we can infer some of the properties of star clusters, the study of which gives insights to stellar evolution. During the “hydrogen burning” phase of very young stars, nuclear reactions in the stellar core fuse hydrogen atoms into helium atoms, and the star is described as being on the “main sequence”.  The length of time a star stays on the main sequence depends on its mass. The higher the stellar mass, the shorter time the star stays on the main sequence.
Hence, among other properties we can estimate the age of the cluster by using their H-R diagrams to judge what proportion of stars have left the main sequence.  The H-R diagrams of open clusters show that most of their stars are on the main sequence, and are therefore at a young evolutional stage. Some of the stars in open cluster M45 are just a few million years old.  In contrast, the H-R diagrams of globular clusters show that a high proportion of stars have left the main sequence and the stellar cores are fusing higher order elements.  Globular cluster M92 appears to have less stars that have left the main sequence than does M2, indicating that M2 is the older of the two.  Some globular clusters appear to more than 12 billion years old.

Acknowledgements
This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.

References
Russell, Henry N (1912).   Relations between the Spectra and Other Characteristics of the Stars  ’ Proceedings of the American Philosophical Society , Oct. - Dec., 1912, Vol. 51,No. 207 (Oct. - Dec., 1912),pp.569-579.https://www.jstor.org/stable/pdf/984021.pdf?refreqid=fastly-default%3A81ac3701b6e4316dd5da365af75ab30f&ab_segments=0%2Fbasic_search_gsv2%2Fcontrol&origin=search-results   Accessed November 5 2022.
Willman, B et al (2010). Willman 1 - a probable dwarf galaxy with an irregular kinematic distribution. https://arxiv.org/pdf/1007.3499.pdf  Accessed November 5 2022.

The clocks went back last weekend so we can see the evening sky a bit earlier and at present with quite mild conditions but I’m sure it will get colder.
 
Observing
The following chart represents the night sky at 11.00pm GMT on the 8th of November and at 10.00pm GMT on the 23rd November. To use the chart, face your southern horizon at the appropriate time and you will see the stars in the chart. If you are observing earlier in the evening just turn eastwards by 15 degrees for every hour before the stated time but objects will be lower in the sky.

Some parts of the chart are the same as last month so a little familiarity should help with navigation. I hope you have been enjoying Cassiopeia at its best, high in the sky going from autumn to winter. Locate Cassiopeia and drop down slightly towards the horizon and to the left of Andromeda you will see the prominent constellation Perseus- representing the mythological Greek hero of the same name, and containing the two bright stars Mirphak and Algol. Again a human shape isn’t obvious but the branch up from the bright star Mirphak is meant to be his right arm and sword while the lower right hand branch is his left arm with the head of Medusa, depicted by Algol, in his left hand. The third branch represents his left leg. Mirphak is a magnitude 1.8 yellow supergiant while Algol has the distinction of being the first eclipsing binary to be observed. In fact it is part of a triple system whose magnitude dips from 2.1 to 3.4 as its components pass in front of each other. With another five stars brighter than magnitude 3 Perseus stands out in the night sky.
            To the left of Perseus lies the unmistakeable Capella, the fourth brightest star in the northern hemisphere at magnitude 0.1 and the brightest star in the constellation Auriga- The Charioteer. Its seven brightest stars form the rough shape of a septagon although its lowest star Alnath is actually in Taurus. Capella is ‘only’ 42 light years from Earth and is a binary system consisting of two yellow giants too close to be resolved in most amateur telescopes.
            As mentioned above, Alnath is in the constellation Taurus- The Bull lying just below Auriga and extending down and towards the right. It is a zodiacal constellation lying between Aries and Gemini. Its brightest star Aldebaran is readily picked out as it is a red giant of magnitude 1 supposedly representing the red eye of the angry bull at the base of the lower horn. The ‘V’ shape to the right of Aldebaran represents the face of the bull and is an open star cluster called the Hyades. The more famous open cluster, the Pleiades or Seven Sisters, is in the constellation Taurus but lies to the northwest of Aldebaran roughly in the direction of Algol. It is worthy of observation in its own right and is one of my favourites for viewing with the unaided eye.
 
Something to look out for
            Mars reaches opposition on the 8th December so it is becoming the planet to watch. It is currently shining at magnitude -1.2 and lies between the tips of the horns in the constellation Taurus and so enables a comparison to be made with the star Aldebaran. Mars just started retrograde motion on the 30th October so it will be moving from east to west against the starry background throughout the month.
            There will be several close approaches of the Moon with different planets, starting with Jupiter on the 4th November. Visible from early evening but best viewed about 9.15pm, 36 degrees above your southern horizon. This is followed by Mars on the 11th November, again visible from early evening but highest in the sky in the early hours. Finally Saturn joins in on the 29th November about 20 degrees above your southern horizon in the early evening.
            For those of you who like to see a ‘shooting star’ the Leonid meteor shower should be at its best around the 18th November from 10.30pm onwards.
Clear skies.