Even with the unaided eye we can make some estimates of the properties of a star against the dark background of the night sky. Perhaps the most obvious observation we can make is that some stars look brighter than others. Look very carefully and you will notice that some stars also reveal a hint of colour.
A Historical Perspective
Most civilisations understood that the stars have different brightness and even now our modern quantitative view is based on early Greek ideas. Over 2,000 years ago Hipparcos catalogued over 850 stars and as part of this process he devised a system for classification of star according to their relative brightness. His scale placed the brightest stars as first magnitude the next brightest as second magnitude and so on.
What little we know about Hipparcos’ early catalogue comes from the Alagest written by Ptolomy some 300 years later by which time his catalogue had been extended to 1,025 stars divided into six magnitude groups from 1st magnitude to 6th magnitude (the latter being just visible to the naked eye). With the discovery of fainter stars using telescopes in the early 1600s required the scale to be extended beyond magnitude 6.
Our Modern Magnitude Scale
In Hipparcos’ system, the larger the numerical value for the magnitude of a star, then the fainter the star is. This is perhaps a little counter-intuitive. Nevertheless, the magnitude scale that we use today retains Hipparcos’ convention, but thanks to British Astronomer Norman Pogson we now have a well defined mathematical definition of magnitude. In the 1850s Pogson proposed a logarithmic scale and established a quantitative basis for the measurement of magnitude. Pogson’s magnitude scale was based on the following rule:
For two objects a magnitude difference of 5 magnitudes corresponded to a factor of exactly 100 in brightness.
Notice that the relationship between magnitude and brightness is logarithmic and not linear. If two stars differ in magnitude by just 1.0 their brightness differs by a factor of approximately 2.51; if the difference is 2.0 their brightness differs by 6.3 (2.51 x 2.51), and so on.
Our modern scale also extends Hiparcos’ system with non-integer and negative numbers; if a star is brighter than 0.0 then its magnitude is expressed as a negative number. Our Sun, for example, has an apparent magnitude of -26.
Apparent and Absolute Magnitude
The magnitude of a star is a measure that is relative to other stars, it has no physical units. The magnitude of a star in the night sky as it appears to an observer is dependent on two factors: how far away it is and how much energy it radiates. For example, consider two household electric light bulbs: side by side and observed from a short distance, a 100W light bulb will appear brighter than a 40W light bulb, but move the 100W light bulb a few hundred yards further away and it appears dimmer than the 40W light bulb.
Astronomers commonly use two types of magnitude– apparent magnitude and absolute magnitude.
The apparent magnitude of a star is the magnitude that an object has on the sky to an observer. The apparent magnitude scale runs from -26 (our Sun) to approaching +30 (stars visible to only the largest telescopes). Some apparent magnitudes are as follows:
| Object | Apparent magnitude |
| Venus | 4.3 |
| Sirius | -1.44 |
| Vega | 0.0 |
| Betelgeuse | 0.45 |
| Alnair | 1.73 |
| Proxima Centauri | 11.09 |
The absolute magnitude of a star is the apparent magnitude that the star would have if it were located at a standard distance of 10 parsecs from the Earth. (1 parsec equals approximately 3.26 light years).
By definition, a star located at exactly 10 parsecs from the Earth will have the same apparent and absolute magnitude. A star that is further away than 10 parsecs will have a fainter apparent magnitude than absolute magnitude and a star that is closer than 10 parsecs will have a brighter apparent magnitude than absolute magnitude.
When comparing absolute magnitudes of stars we are effectively taking one of two variables out of the comparison – the stellar distances. The absolute magnitude of a star is therefore a measure of its Luminosity – the total amount of energy radiated by the star per second. The luminosity is a quantity that depends on the star itself, not on how far away it is. Luminosity can tell us about the internal physics of the star and is a more important quantity than the apparent brightness.
Our Sun, m=-26.5, has an absolute magnitude of M=4.85. Note that the convention is to use upper case when referring to absolute magnitudes (e.g. M=1.0) and lower case when referring to apparent magnitude (e.g. m=2.0).
How do we Measure a Star’s Absolute Magnitude?
The answer is that you can’t – not directly – but it can be determined if we know other characteristics of the star. There is a very important relationship between absolute magnitude (M), apparent magnitude (m) and distance (d) – this relationship is called the distance modulus:
m – M = 5 x log(d) – 5
If we know a the absolute magnitude of a star and its apparent magnitude, we can then calculate its distance using the distance modulus formula. Similarly, if we know the apparent magnitude of a star AND its distance we can then calculate its absolute magnitude.
For example, if a star is a distance of 10pc then:
m – M = 5 x log(10) – 5 = 5 x 1 – 5 = 0; therefore m = M
At 10pc the distance modulus is zero. This fits exactly with the definition of absolute magnitude as stared above – i.e. that for a star at a distance of 10pc absolute and apparent magnitudes are the same. If however, d = 3.16pc then:
m – M = 5 x log(3.16) – 5 = 2.5 – 5 = -2.5
In other words the star is closer than 10pc and is shown by the distance modulus to have a brighter apparent magnitude than absolute magnitude. If the apparent magnitude of the star is m = 2.5 then the absolute magnitude of the star is M = 5.
More About Luminosity
Luminosity is the total amount of energy radiated by the star per second. A star can be luminous because it is hot, because it is large or both. The amount of energy radiating from every square metre of a stars surface is proportional to its temperature raised to the fourth power (Stefan-Boltzman Law ); this means that the luminosity of a star increases massively with only small increases in its temperature.
The luminosity of a bigger star is larger than a smaller star at the same temperature because it has a larger surface area. Therefore a small, hot star can have the same luminosity as a large, cool star. If the luminosity remains the same, then the larger a star is the lower its temperature must be.
This is a very important relationship because if we know the apparent brightness of a star as well as its temperature, and distance, we can then determine its size.
What is Limiting Magnitude?
In its widest sense, limiting magnitude is the faintest apparent magnitude of an object that is detectable or detected by a given instrument – such as a 4”, an 8” SCT or even a large observatory telescope. However, in amateur astronomy, limiting magnitude is frequently used to refer to the faintest stars that can be seen with the unaided eye near the zenith on a clear moonless night. It is most often used as an indicator of ‘sky brightness’ and light pollution. Light polluted areas such as towns and cities generally have a lower limiting magnitude than remote or high altitude areas.
Try this as an exercise: on a moonless night identify the dimmest stars that you can observe from your back garden and maybe a darker site outside of your town or village. Look up the magnitudes of these stars on a star chart of planetarium to determine the limiting magnitude of these locations. Try this on different nights; limiting magnitude can vary according to atmospheric conditions (e.g. humidity). In ideal conditions, the limiting magnitude of the eye is about 6.5.
Herein lies a danger for the unwary beginner; some amateur astronomers will talk endlessly about their favourite dark observing sites their limiting magnitudes. You have been warned!
Bolometric Magnitude
Strictly speaking, when stating a star’s luminosity the pass band that the star’s luminosity was measured in should be stated because magnitude varies with wavelength. A commonly used photometric system is the UBV system – The wavelengths where each filter transmits most light are respectively: U = 360 nm, B = 440 nm, and V = 550 nm. (V approximates to the human eye). Another type of magnitude of particular interest to theorists is the bolometric magnitude, which is a magnitude based on measurement throughout the entire spectrum.
Conclusion
This article has shown how some basic properties of stars – apparent magnitude, absolute magnitude, distance, temperature and size – are all interrelated and how knowing some of these properties we can learn much more.
In future articles we will cover some of the other properties of stars used by Astronomers to learn more about stars and the stellar lifecycle. This article finishes with a wonderful example I found of some ‘famous last words”:
“We can never by any means investigate their chemical composition… The positive knowledge we can have of the stars is limited solely to their geometrical and mechanical properties” (Auguste Comte, Cours de Philosophie Positive, 1842)
Thankfully for us, August Comte was very wrong!