Classification by Luminosity

Luminosity is the rate of electromagnetic radiation emitted by a body per unit of time. It may be a misleading term since we tend to think of luminosity and similar physical terms as quantities associated with the visible radiation we perceive, whereas it is a measure of the whole spectrum of electromagnetic radiation.

Therefore, we find that in order to appropriately characterise the properties of a star, we need to take measurements in every possible frequency, not just the ones we can perceive with our eyes.

Luminosity and distance

When we consider luminosity, we have to take into account that all the measurements we perform are dependent on the place we occupy in the universe. The reason for this is that astronomical objects like stars radiate in every direction, and their waves spread very quickly, so we only perceive a fraction of the whole picture (which we can then extrapolate).

Luminosity and extinction

Space is not as empty as it seems, especially when we consider the large distances appearing in astronomical settings. Certain structures and objects like space dust appear quite often and interfere with the detection of radiation emitted by bodies. This leads to a phenomenon called ‘extinction’, which is nothing more than the loss of intensity of electromagnetic radiation.

This phenomenon, in astronomical settings, affects more prominently high-energy radiation, such as gamma radiation or x-ray radiation, than low-energy radiation like radio waves or infrared radiation.

Extinction and the distance from the earth (or wherever we have placed our telescope) are two factors that make our measurements highly dependent on the place we occupy in the universe. However, by means of further direct and indirect measurements and calculations, we can accurately estimate many of these quantities. Although luminosity, as perceived by us, is a rather subjective quantity, it is still useful and is a measure of the intensity we perceive from the earth.

Classifications by luminosity

Classifications by luminosity are systems of the categorisation of stars and other astronomical bodies that attend to the intensity of the electromagnetic radiation they emit. We are now going to study a certain classification of stars by luminosity, and then we will briefly mention the stellar spectral classification, which is related to classifications according to luminosity.

The luminosity classification of Hipparchus

Already in the 1st century BCE, Hipparchus, a Greek astronomer, made an attempt to classify stars by the brightness with which he perceived them at sight. He used a system of numbers, ranging from one to six, which he denominated magnitude. The brightest stars were assigned a magnitude of 1, while the darkest stars, the ones he could barely see, were assigned a magnitude of 6.

The actual concept of luminosity

Due to several theories and advances, we now have several models for the luminosity of stars and other astronomical objects. Most of these models are based on the concept of black body radiation, which assumes the perfect emission properties of objects. According to this assumption, which is approximately fulfilled by stars, luminosity L is defined as follows:

\[L = \sigma \cdot A \cdot T^4\]

Here, σ is the Stefan Boltzmann constant (with an approximate value of 5.67·10^-8 W/m²K⁴), A the area of the emitting body, and T its temperature in Kelvin.

Therefore, we find that the luminosity of stars is related to their surface size (and, hence, their volume) and temperature. A big star with a small temperature may have the same luminosity as a small star with a higher temperature.

However, luminosity is a very difficult quantity to measure as we need to measure the emission in every possible direction. The closer we measure to the surface of the emitting body, the less relevant will be the effects of extinction. In addition, we have to remember that the perceived luminosity also depends on the distance to the emitting object.

Yerkes classification

The following table has the Yerkes luminosity classification chart, which assigns each star a number based on their luminosity. It is the most widely used classification by luminosity for stars.

Magnitudes and luminosity

To prevent the difficulties of measuring or calculating luminosity, we usually turn to the concepts of absolute and apparent magnitude. These are logarithmic scales for the luminosity of astronomical objects measured at 10 parsecs from the object (absolute magnitude) or a fixed place like the earth (apparent magnitude). By having a common reference point, it is easier to match data. One parsec equals 3.09·10¹⁶ m.

However, the problem of extinction is still present, and most calculations are done by using estimations for its effect. Telescopes cannot reach almost any astronomical object to take measurements at a distance of 10 parsecs, so these quantities have complex models that allow us to estimate them accurately.

The relationship between luminosity and spectral classifications

If we fix area A in the luminosity formula, we just have a dependence on the temperature of the star or astronomical object. Hence, under the assumption that stars emit as black bodies, using quantities such as the absolute magnitude allows us to consider only their temperature.

In addition, there is a phenomenon called Wien’s law, which takes into consideration how a black body emits radiation at a certain temperature. It turns out that the intensity of emission of frequencies depends on the temperature.

Since stars emit in all frequencies (with different intensities depending on their temperature), we observe a dependence of their colour on their temperature. This gives rise to the concept of stellar spectral classification.

Although we cannot here explore how stellar spectral classifications work, it is important to be aware of the system and that temperature is correlated with luminosity, as illustrated by diagrams such as the Hertzsprung-Russell diagram (see also the table below). The luminosity of a star varies throughout its life and depends on certain phases like pulsating phases, the giant phase, or the white dwarf phase.

Computations with classifications by luminosity

Sirius is the brightest star that can be seen from the earth. It has a luminosity on its surface of 25.4 times the luminosity of the sun (L₀), a radius of 1.711 times the radius of the sun (r₀), and is 8.61 light-years away from the earth. Sirius is a main-sequence star (V).

We can compute the temperature for both stars, assuming they behave like black bodies. Indeed, knowing that the luminosity of the sun has an approximate value of 3.83·10²⁶ W and the radius of the sun has an approximate value of 6.96·10⁸ m, we can use the following formula, which comes from the formula of luminosity, using the fact that stars are spherical bodies, so we can easily compute their surface area:

\[T = \sqrt[4] {\frac{L}{4 \cdot \pi \cdot \sigma \cdot r^2}}\]

This yields a temperature of 9904 K for Sirius and 3673 K for Antares. These values are very close to the measured ones, which are 9940 K and 3660 K, respectively. Since Sirius has a higher temperature than Antares, its colour is much bluer.

We finally turn to compute the luminosity of these two stars as perceived on the earth. The formula is:

\[L_p = L \cdot \frac{d^2}{r^2}\]

Here, r is the radius of the star, while d is the distance of observation, which, in this case, is the distance to the earth. L is the luminosity of the star, and L_p is the luminosity observed from a distance d, which is lower due to the spherical spreading of electromagnetic radiation.

Using the values, we get observed luminosities per area of 1.14·10^-7 W/m² and 8.54·10^-8 W/m², respectively, which explains why Sirius is brighter than Antares as seen from the earth. This is related to the concept of magnitude.