Stars - from what we see to what we measure by Gordon Dennis

Abstract
In this Blog, I examine the properties of four well known stars – the Sun, Betelgeuse, Sirius, and Canopus. Despite the distances to the last three stars, we do know quite a lot about them.  I examine how we can measure stellar characteristics, and equally important, what the uncertainties are in such measurement.  In historical context, I identify the key astronomers and physicists whose discoveries have led to our present-day understanding.

Introduction
We can't visit other stars to make measurements.  Any notion of inter-stellar travel is way beyond any technology that we possess today.  Even the closest star to us, Proxima Centauri is approximately 4.25 light-years (lyr) distant.  And yet we do know a lot about stars. In this Blog I will discuss some ways in which this has been achieved, and also see some of the limitations and uncertainties surrounding our observations and measurements.
The key discoveries underlying what we now know were made over a century ago.  Today, astronomers and physicists are still investigating ways to improve the accuracy and precision of measurement in these areas.
In this Blog, I describe two key stellar parameters that enable us to derive other stellar parameters.  These parameters are the luminosity of stars and the surface temperature of stars (known as the photospheric temperature).  I show how we can use these to calculate the size of a star and the radiation environment the stars produce.  Finally, I discuss how the radiation would affect any Earth-like lifeforms on exoplanets in orbit around the star.
From here on, we consider a small sample of four stars:

• The Sun
• Betelgeuse,  orionus, a red super-giant in Orion (apparent magnitude +0.50)
• Canopus,  carinae in the southern constellation of Carina and the second-brightest star in the sky (apparent magnitude −0.74).
• Sirius,  Canis Majoris in Canis Major, the brightest star in the sky (apparent magnitude -1.45)

Figure 1 shows a star map depicting the location of our four-star sample. This view is not fully visible from the  UK because Canopus is below the  hoirzon at our latitudes.  In the northern hemisphere, Canopus is visible as far north as Florida.

Spoiler alert: there are some equations in here (GCSE level), but don't worry we're just going to use them, not derive them from scratch.

Luminosity of stars
Let's ask "how bright is a star?" Scientifically, luminosity is a measure of the power of the radiation emitted by a star. The SI unit of luminosity is the Watt. The luminosity of the Sun is called one solar luminosity, quantitatively:

Luminosity of other stars is often expressed in units of solar luminosity as we shall see later.
We distinguish luminosity, which is independent of its distance from us, and the brightness we actually see which depends very much on how far away the star is because of Newton's inverse square law:

By measuring the brightness that we observe, we can work back from there to determine the star's intrinsic luminosity, so long as we know the distance. 
Figure 2 below shows the intrinsic luminosity of stars in our four-star sample:

But, how precise can we be? Let's take the case of the star Betelgeuse in Orion. The luminosity of Betelgeuse is stated as:

Perhaps this is a shorthand way of saying "we don't know the luminosity of Betelgeuse to high precision". The reason is that we don't know the distance to Betelgeuse to high precision.
We can see both Betelgeuse and Canopus are much more luminous than Sirius, so how can Sirius be the brightest star in the sky? The answer again is distance. Distances of our sample are shown in Figure 3 below:

We see from this plot that Betelgeuse is by far the most distant of our 4-star sample, while Sirius is the closest star to us, apart from the Sun of course. Sirius is only 8.6 lyr distant whereas Betelgeuse is 548 lyr distant. Sirius appears so bright to us because it is close to us.


But there are uncertainties in distance measurement. Being further distant makes the distance measurement less precise. We can see in the chart above that the error bars on Sirius' distance are very small. The error bars on Canopus' distance are wider. The error bars on Betelgeuse's distance are much wider – which means our distance measurement is far less precise (Joyce et al, 2020). In fact, Betelgeuse is just about as far distant as we can measure using the parallax method.

How large are stars?
The Stefan–Boltzmann Law is named for Slovene physicist Josef Stephan (1835–1893) and Austrian physicist, Ludwig Boltsmann (1844-1906) – see Figure 4.

The Stefan–Boltzmann Law describes the power radiated by a body that absorbs all radiation that falls on its surface in terms of its temperature. This type of object is called a black body. Any object at a stable temperature above the absolute zero is a black body.
We assume that stars are examples of a black body, and that they are spheres; as a first approximation these assumptions are true. With those assumptions, we may use Stefan–Boltzmann law, in the form of the equation:

where the Stephan-Bolzmann constant, σ = 5.67 *10-8J s-1m-2K-4,   is the stellar radius, which is the size of the star that we want to determine; and T is the temperature of the star’s surface.   
Because the Sun is so close, we can measure its luminosity directly. We can also measure its surface temperature.  Re-arranging the Stephan-Bolzmann equation, we get:

Which enables us to derive the Solar radius. Note also that this measure of Solar radius can be verified by other techniques.  The solar parameters we need are:

Remember we said that parameters of distant stars are often expressed in terms of Solar units.  Why would we want to do this?  Suppose that we have determined luminosity of a distant star, L*, and we have measured the star’s photospheric temperature, T*. using spectroscopy.
Now, using the Stephan-Bolzmann equation for the Sun and for the remote star respectively, and dividing one by the other, we see all the constants cancel out:

We now re-arrange this equation to give us an equation for the remote star’s radius:

This equation is in terms of things we can measure.  At first glance, it may look a bit ugly, but it’s easily solvable with a scientific calculator (or Excel, our whatever your preference may be), and gives us the data shown in Figure 5 for our four-star sample:

But what about the precision of our measurements?  Returning to the case of Betelgeuse, which was the first star ever to be observed as a disk (as opposed to just a point of light). Here we come to some further uncertainties.  Looking at the Figure 6 below, we can see that Betelgeuse is not a black body – if it was, it would appear with uniform brightness over its entire surface.  Also, Betelgeuse is variable in brightness over quite short periods.   Figure 6 shows the recent and unusual dimming of Betelgeuse over a period of ~ 1 year

However, like many red supergiants, Betelgeuse has more than one periodicity of variability (Joyce et al, 2020).  This variability over a period of approximately 14 years is shown by the observed light curve in Figure 7 below:

We can see two things in Figure 7.  First there is a long-term pulsation in brightness with a period of approximately 420 days, indicating Betelgeuse is in an unstable state.  Second, superimposed on the long-term cycle, there was an unusual dip in brightness in 2019 (see Figure 6).
Thirdly, it's very difficult to measure the radius of Betelgeuse by astrometry because what we see is a fuzzy view of the star's atmosphere. There is no sharp looking outer limit such as we see at the top of the atmospheres of gas giant planets in the Solar system. We must accept that we don't know the stellar parameters of Betelgeuse with high precision.
Allowing for these uncertainties, we can say that Betelgeuse is approximately 3 orders of magnitude larger than the Sun - in this case, orders of magnitude being in the mathematical sense rather than astronomical magnitudes. Canopus is ≈2 orders of magnitude larger than the Sun. Sirius is of similar size to the Sun (1.71R⊙).

What part of the spectrum do stars radiate at?
If you've read this far,  you'll appreciate that nowhere have we yet mentioned wavelengths of the radiation. The wavelengths at which a star radiates is a key parameter in determining whether any of its planets might harbour life. For example, if an exoplanet is bombarded with ultra-violet radiation, it certainly won't harbour life anything like human beings.
Wien’s Law is named after the German Physicist Wilhelm Wien (1864–1928), who published his Law in 1893 and won the Nobel Prize for Physics in 1911.

Wien’s Law shows that objects of different temperatures emit radiation that peaks at different wavelengths. The peak wavelength is given by the empirical equation:

Once again assuming the stars in our 4-star sample to be black bodies, the radiation curve at various temperatures has a form shown in Figure 9 below:

Examining the green curve, which shows the peak wavelength of the Sun, we see that it peaks in the visible light spectrum. The Sun does also radiate in the near ultra-violet but, fortunately for us, Earth's atmosphere blocks most ultra-violet radiation.
In contrast, examining the black curve, we see that the peak wavelength of Sirius is in the ultra-violet (UV). Sirius is a binary system (Bond et al 2017).  Sirius A is a spectral class A star, and its companion Sirius B is a white dwarf. No planetary system has been found orbiting around either or both stars. If there are any planets in the Sirius system, the UV radiation from Sirius A would mean that any lifeforms on those planets would of necessity have evolved differently to life on Earth

Energy radiated by stars
We can get a good idea of how damaging the UV radiation from Sirius A is. If we look at the black (Sirius) curve, and compare it to the green (Sun) curve, we can see straight away that the area under the Sirius curve is dramatically larger than that under the Sun curve.  Also, there is a lot more area under the Sirius curve in UV wavebands than in the case of the Sun. Simply put, there is not only a lot more energy being radiated by Sirius, but also a greater proportion of Sirius’ energy is radiated as UV.
Starting from Wein’s Law, the peak emitted wavelength of Sirius A is:

We quantify the radiation by starting from the Planck-Einstein equation, which gives us the energy per photon:

Where  is Planck's constant,  is the speed of light in vacuum,  is the frequency of the radiation, and  is the corresponding wavelength of the radiation.
Hence, the photon energy of Sirius A at the peak wavelength is:

If we do the same calculation for the Sun, the peak photon energy

In other words, every photon radiated from Sirius A at its peak wavelength carries almost twice the energy of a photon radiated by the Sun at its peak wavelength, as we see in the Figure 10 below:

How dangerous to Earthly lifeforms would this be? UV radiation is divided into three wave-bands:

Sirius-A's radiation is mainly in the UV-B waveband, and an appreciable amount is in the UV-C waveband.  Any advanced Earth-like lifeforms would be at high risk of skin cancer – unless those lifeforms had evolved differently (for example so that UV was actually required for their life), or their planet's atmosphere blocked the UV-B out, as does Earth's atmosphere.

Conclusion
Figure 11 shows a summary of the characteristics of our four-star sample.  This indicates:

Betelgeuse is both by far the most luminous star in the sample, and also by far the largest star.  Even the next largest star in the sample, Canopus, is much smaller than Betelgeuse.
The Sun is comparable in size to Sirius, but has a much lower photospheric temperature.  The Sun does radiate UV, but not nearly to the extent as Sirius does.  No planets have been discovered in the Sirius system but if planets did exist, the UV radiation created by Sirius if potentially harmful to Earth-like lifeforms.
Due to its size and its high photospheric temperature, Canopus has a much higher luminosity than either the Sun or Sirius.

The most well-known and important diagram in astronomy is the Hertzsprung-Russel diagram, which helps us study stellar evolution, whereas the diagrams in Figure 11 show stellar parameters of out four-star sample as they appear today. I won't go into H-R diagrams here as I've already exceeded my word count - maybe in a future Blog.
Thanks for your interest.

Acknowledgement
The author gratefully acknowledges Dr. Ovidiu Borchin and Mr. Bob Merritt for reviewing this paper.  Their suggestions have resulted in significant improvements to this work.

References
Bond, H et al (2017). The Sirius System and Its Astrophysical Puzzles: Hubble Space Telescope and Ground-based Astrometry.  ApJ, , 840:70 (17pp), 2017 May 10.  https://iopscience.iop.org/article/10.3847/1538-4357/aa6af8/pdf   Accessed July 11, 2022
Joyce, M et al (2020). Standing on the Shoulders of Giants: New Mass and Distance Estimates for Betelgeuse through Combined Evolutionary, Asteroseismic, and Hydrodynamic Simulations with MESA. ApJ, 902:63 (25pp), 2020 October 10.
https://iopscience.iop.org/article/10.3847/1538-4357/abb8db/pdf .   Accessed July 11, 2022