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# Hertzsprung-Russell Diagrams

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Astrophysics relies on observation because most of the objects and phenomena it studies are not available in a laboratory. They also involve huge amounts of energy, matter, speed, distance, etc., which make it impossible for us to perform measurements. For instance, while we have plenty of theoretical models for the inner structure of the sun, we have not been able to perform direct measurements because of the extreme temperatures reached by stars.

These limitations already inspired ancient astronomers to adopt a strategy of statistical inference. If one collects enough data for the same phenomenon by using different characterisations that are measurable from the earth, statistics could provide a first approach to understanding the basic characteristics of that phenomenon. Ancient astronomers, who had not yet developed telescopes, would study the brightness of stars as seen by their eyes and record their measurements related to the position of these stars in the sky. This provided a first approach to the question of whether the sky or cosmos is homogeneous (equal or almost equal in any direction).

Modern astronomers and astrophysicists, by contrast, have a large number of advanced devices that have improved the quality of measurements made from the earth (or the surface of close planets like Mars). This, along with development in physics itself, has allowed them to construct more sophisticated statistical models that capture some relevant information of astronomical phenomena. The Hertzsprung-Russell diagram is one of the most widely used. It manages to capture the stages of stellar evolution by classifying thousands of stars according to their luminosity and surface temperature.

## What is the Hertzsprung-Russell diagram?

The Hertzsprung-Russell diagram is a graphical two-dimensional representation of every known star according to its luminosity and surface temperature. The reason for choosing these two variables will be explored below.

Figure 1. Hertzsprung-Russell diagram. Source: ESO (CC BY 4.0).

### Luminosity

Luminosity is the measure of the total electromagnetic power emitted by an object. It is measured in Watts (W), which is defined as Joules (energy unit, J) per second.

Measuring luminosity is not a simple task due to the following factors:

• Electromagnetic radiation spreads over space as it propagates in the same way as a wave caused by a stone thrown into a lake. Since we are usually very far away from the objects whose luminosity we are measuring, we are only receiving a portion of the total radiated energy, which we have to take into account.
• The transmission of electromagnetic radiation is not perfect. There are different structures between the emitting body and the measuring device that may partially (or even totally) dissipate the electromagnetic radiation, which would render the measurements incorrect.
• Luminosity can be measured for every possible frequency. We should, therefore, not restrict ourselves to the visible range of the electromagnetic spectrum that we can perceive with our eyes because, depending on the characteristics of the star, the luminosity measured in other regions can be high and thus relevant.

However, if we can estimate the total luminosity from the spread data we receive (by knowing the distance between the measuring device and the star), model the dissipation caused by astronomical structures, and measure in all electromagnetic frequencies, we can accurately know the radiating power of stars without being near them. These three requirements are achieved with modern techniques and technologies, and there is constant progress regarding the precision of these methods.

In connection with the Hertzsprung-Russell diagram, we are going to consider how luminosity is relevant for characterising a star. It is a measure of its radiated power, which should be related to its brightness if we are looking from a spatial point in its vicinity (remember that the brightness we perceive is strongly affected by spreading and dissipation). So, in a first approximation, a star’s luminosity is related to:

• Nuclear power generated inside the star: the bigger the power, the brighter the star.
• The star’s radius: the bigger the star, the larger the surface that emits light, which is equivalent to a higher amount of emitted energy per unit of time.

If we have two stars with the same reactions happening at the same pace, the bigger one will have a higher luminosity. On the other hand, if we have two stars of the same size but with different nuclear reactions generating power, the one whose nuclear reactions generate more energy per unit of time will have a higher luminosity. For almost all types of stars, it can be assumed that the nuclear reactions are similar and that we can take the luminosity as an indirect measurement of the star’s radius.

### Wien’s law and the chromaticity-temperature relationship

Let us briefly consider why the colour of a star (its chromaticity) is related to its surface temperature. Quite remarkably, black bodies have many thermodynamic properties that are almost exactly followed by stars, which suggests that the approximation is also almost true for these astronomical bodies.

Black body is the name given to a perfectly emitting and absorbing system.

Emitting bodies, such as stars or black bodies, emit thermal energy in small amounts because most of the energy leaves the body as electromagnetic radiation created by thermal processes within the emitting body. This suggests that for emitting bodies, we can find a relation between the surface temperature (where the emission happens) and some characteristics of the emitted electromagnetic radiation. The quantity related to the surface temperature in this setting is the electromagnetic frequency of the radiation (or the wavelength, which is related to the inverse of the frequency).

For electromagnetic radiation, we have a classification of types of waves in terms of their frequencies (x-rays, radio waves, visible radiation, etc.). In the visible region, frequencies correspond to what we perceive as colours. The higher frequencies in the visible region correspond to blue and violet colours, while the lower ones correspond to red and orange colours.

Figure 2. The electromagnetic spectrum and its scales. Source: Inductiveload, NASA, Wikimedia Commons (CC BY-SA 3.0).

There is a law for black bodies that determines which frequencies are emitted more intensely for a specific temperature. Since stars are approximated with great accuracy by black bodies, we can apply this law, which is phenomenologically described in the following diagram, also to them.

Figure 3. Black body emission for different temperatures. Source: MikeRun, Wikimedia Commons (CC BY-SA 2.5).

As we can see, at higher temperatures, a body emits electromagnetic radiation with a shorter wavelength (higher frequency or ‘bluer’), while at lower temperatures, the intensity of emission shifts towards longer wavelengths (lower frequencies or ‘redder’). This is the reason why in the Hertzsprung-Russell diagram shown in Figure 1 the colours are included as well: they are directly related to the surface temperature of the stars. Hot stars are blue, and cold stars are red.

The spectrum of emission thus determines with great accuracy the thermal properties of the body under study. By studying the intensity of the incoming radiation in all frequencies, we can come to conclusions about both the star’s luminosity and its temperature. However, it is important to note that, although they come from the same data, they are unrelated quantities. The luminosity (with the necessary corrections applied) is related to the sum of all the intensities at all frequencies, while the temperature is related to how these frequencies are distributed.

On the one hand, this means that if the shape of the curve of the graphic intensity of emission vs wavelength is the same, the two stars are at the same temperature. However, while they may have the same shape, one has double the height compared with the other, which indicates that the first one has double the luminosity. That is, if they are at the same distance and in the same region, as otherwise we would have to take into account the dissipation and spreading. On the other hand, if there are two diagrams whose total sum is the same while the shapes are completely different, we would have two stars with the same luminosity (given the previous assumptions) but with different temperatures.

## How is the Hertzsprung-Russell diagram used?

Rather than developing a complex model to determine how the surface temperature and luminosity are correlated for all types of stars, we can use the diagram after having collected enough data to make predictions about the nature of these quantities. It turns out that, due to the processes happening inside stars, these two quantities, which are not constant throughout the life of stars, determine their stage of life. The Hertzsprung-Russell diagram can thus be used as a visual way of representing the life of stars.

### Main sequence

The main sequence is the stage in which stars spend most of their lives. The initial characteristics of a star, such as its mass, determine where in the main sequence it starts, from where it then slowly evolves down and right-wards, which is to say that stars tend to decrease their temperature and luminosity as they age.

### The giants and supergiants branches

Stars do not always remain in the main sequence. Due to internal processes, they can change their luminosity and/or surface temperature drastically and become giants or supergiants (these terms indicate that their radius increases). Stars with the highest masses become supergiants, while those with intermediate or low masses become giants. Our sun, for instance, is close to the point where it will evolve to become a giant in 4-5 billion years.

### The white dwarfs branch

The fact that the branch associated with white dwarfs is disconnected from the rest of the diagram indicates that some sudden and drastic event needs to happen for stars in the main sequence to reach this state. At the end of its life, a star either becomes a black hole or explodes in a supernova, which leaves no emitting body behind that whose luminosity and temperature could be studied. However, after certain supernovae corresponding to low-mass stars, astronomical bodies remain that look like a regular star but have a much lower luminosity and surface temperature: white dwarfs.

## Hertzsprung-Russell Diagrams - Key takeaways

• Astrophysics relies on the collection of data and the statistical processing of the information gathered.
• Relevant quantities of stars, such as their size or temperature, can be estimated through certain models and the measuring of the electromagnetic properties of the spectrum of emission.
• The Hertzsprung-Russell diagram is a representation of stars according to their luminosity and surface temperature. The pattern formed after the collection of significant amounts of data allows us to study the evolution of stars.
• The Hertzsprung-Russell diagram has four main regions: the main sequence (where stars spend most of their lives), the giants and supergiants branches, and the white dwarfs branch, which signals the final stage of life of some of the stars of the main sequence.

It is a diagram in which stars are classified according to their luminosity and surface temperature. It allows to track the evolution of stars.

In the Hertzsprung-Russell diagram, there are four main regions: the main sequence, the giants and supergiants branches, and the white dwarfs branch.

The luminosity gives a very accurate measure of the radius of stars under reasonable assumptions. The higher the luminosity, the larger the radius.

The main sequence, the giants’ branch(es) and the white dwarf branch.

In the main sequence.

## Final Hertzsprung-Russell Diagrams Quiz

Question

Select the correct statement:

Astrophysics is a science that uses a lot of statistical techniques.

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Question

Select the correct statement:

Luminosity can be estimated from the measured radiation intensity, but one needs to take into account other factors.

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Question

Select the correct statement:

There is a well-known relationship between the temperature of a black body and its emission spectrum.

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Question

Select the correct statement:

The luminosity and temperature of stars change throughout their lives.

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Question

Select the correct statement:

There are four main regions in the Hertzsprung-Russell diagram: the main sequence, the giant and supergiant branches, and the white dwarfs branch.

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Question

Are stars accurately described as black bodies?

Yes, they are.

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Question

Can the measured luminosity be affected by astronomical structures?

Yes, it can be dissipated by them.

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Question

Under assumptions of similar nuclear reactions, what is the main quantity related to luminosity?

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Question

Are hotter stars bluer or redder?

Hotter stars are bluer.

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Question

Are colder stars bluer or redder?

Colder stars are redder.

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Question

Towards where do stars evolve in the main sequence?

Towards the lower luminosity and lower temperature regions.

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Question

What determines whether a star will enter the supergiants or the giants branch?

Its mass.

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Question

What is the next stage of life of the sun?

To grow in size and become a red giant.

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Question

How do white dwarfs originate?

They are the remnant objects of some supernovae.

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Question

To which type of bodies does the relationship between the spectrum of emission and the temperature apply?

Black bodies.

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