## The brightness of stars

Looking at the stars on a dark, clear night one of the most obvious features is that they are of different brightness (Figure 1). Some are bright while others are at the limit of naked eye visibility and everything in between. Scanning the sky with a pair of binoculars or a telescope will bring many fainter stars into view.  The larger the aperture of the telescope or binoculars the fainter the stars you can see. Having a system to describe a star’s brightness is useful for many reasons in astronomy, including the scientific study of variable stars and describing the brightness of a new object in the sky such as a nova, supernova or comet.

This tutorial introduces the stellar magnitude scale which is used to describe the brightness of the stars and a method is described which predicts the faintest stars any given telescope will show. The concepts of apparent and absolute magnitude are introduced which allow meaningful comparisons of the stars’ true brightness to be made. Finally there is a discussion of how stars can have different brightness in different colours.

##### The magnitude scale

One of the first people to attempt to categorise the brightness of stars was the Greek astronomer Hipparchus in the second century BC. He divided the stars into the six groups that we now call magnitudes (strictly the correct term is apparent magnitude but magnitude is generally used). The brightest ones he called stars of the first magnitude. Those of the second magnitude were fainter and so on down to the faintest stars visible to the naked eye which he called sixth magnitude. The key point to remember is that the larger the numerical magnitude the fainter the star.

This is obviously quite a coarse measure and in the 19th century it was refined into the numerical system we use today. We retain the concept of six magnitudes but add precision by using continuous numerical values. So for example the bright star Rigel, in the constellation of Orion has a magnitude of 0.18, while the star Procyon, not far away in Canis Minor has a magnitude of 0.40. Both are first magnitude stars, but Rigel appears brighter than Procyon. The scale continues beyond the six magnitudes visible to the naked eye to fainter stars with seventh, eighth, ninth magnitudes and so on. A few stars and some solar system objects can appear brighter than magnitude 0.0 and are given negative magnitudes.

At the other end of the scale, fainter and fainter objects have higher and higher magnitudes and the limit is continually being pushed back with the building of larger telescopes and more sensitive detectors.

There are 21 first magnitude stars, 15 of which are visible from at least some part of the UK. The table below, extracted from data in the BAA Handbook, lists these 21 stars:

Star Designation Magnitude
Sirius Alpha Canis Majoris -1.44
Canopus Alpha Carinae -0.62
Arcturus Alpha Boötis -0.05
Alpha Centauri Alpha Centauri -0.01
Vega Alpha Lyrae 0.03
Capella Alpha Aurigae 0.08
Rigel Beta Orionis 0.18
Procyon Alpha Canis Minoris 0.40
Betelgeuse Alpha Orionis 0.45 (Variable)
Achernar Alpha Eridani 0.45
Beta Centauri Beta Centauri 0.61
Altair Alpha Aquilae 0.76
Acrux Alpha Crucis 0.77
Aldebaran Alpha Tauri 0.87
Spica Alpha Virginis 0.98
Antares Alpha Scorpii 1.06 (Variable)
Pollux Beta Geminorum 1.16
Fomalhaut Alpha Piscis Austrini 1.17
Deneb Alpha Cygni 1.25
Mimosa Beta Crucis 1.25
Regulus Alpha Leonis 1.36

Generally, stars with magnitudes down to 1.5 are considered to be of the first magnitude. Those from 1.5 to 2.5 are second magnitude, 2.5 to 3.5 third magnitude and so on.

It is not just stars which can be allocated magnitude values for brightness, planets too, and other bodies which either emit or reflect light, including comets, asteroids, moons, and more distant entities including nebulae, star clusters and galaxies.

As the planets in our solar system orbit the Sun their distances from both the Sun and the Earth change. Because of this, the apparent brightness and hence magnitudes of the planets varies as viewed from Earth. The table of Solar System objects below shows the magnitudes of our nearest neighbours when they are at their brightest.

Object Magnitude
Sun -27
Full Moon -13
Mercury -1.8
Venus -4.4
Mars -2.8
Jupiter -2.5
Saturn -0.2
Uranus 5.7
Neptune 7.6
Pluto 13.7

The planets out to Uranus are all visible to the naked eye at some time although Uranus will require a clear, dark sky free from light pollution and a keen eye.

##### Magnitude mathematics

This shaded section delves a little more deeply into the mathematics of the magnitude scale. If you wish you can skip this and go straight to the next section.

The magnitude scale is what is known as a geometric progression and this can often be confusing, particularly for beginners. A geometric progression is one where each number is the previous one multiplied by a fixed amount. For example 1, 2, 4, 8, 16, 32, 64 is a progression where each number is twice the previous one.

With the stellar magnitude scale however we do not use a simple number like two to multiply each step instead we use approximately 2.512. Now, this may seem a strange number to use but there is a good reason for it. If you multiply it by itself five times you get one hundred.

This means that:

• If one star is magnitude 1 and another star magnitude 2, then the first star is about 2.512 times as bright as the second.
• Stars differing by two magnitudes are a factor of 2.512 x 2.512 = 6.310 apart and so one is a bit more than six times brighter or fainter than the other.
• If two stars differ by five magnitudes, for example first and sixth magnitude, then one is 100 times brighter or fainter than the other.
• The following table shows how a difference in magnitudes translates to a difference in brightness. So, for example a magnitude 2.5 star is nearly 40 times brighter than one of 6.5 (difference 4.0 magnitudes)
Magnitude
Difference
Brightness
Difference
1 2.512x
2 6.310x
3 15.849x
4 39.811x
5 100.000x
##### How faint can you see?

The faintest magnitude visible to the naked eye is generally given as around 6.5 in a dark sky with normal eyesight (corrected with glasses or contact lenses if required). Some fortunate individuals seem to be able to surpass this and reach perhaps magnitude seven or slightly fainter but these are very much the exception.

Sadly, for many of us a dark sky is the exception rather than the rule. Those living in or near to cities or other sources of light pollution will be unable to see sixth magnitude stars with the naked eye. The glow in the sky from light pollution will wash out the faintest stars (Figure 2). The term “limiting magnitude” is used to describe the faintest stars which can be seen from any given location be it with the naked eye, binoculars or a telescope. From the centre of a city maybe only the Moon and the brightest planets are visible, but as you move away from the city and as the light pollution lessens, fainter and fainter objects become visible to the naked eye. The BAA Commission for Dark Skies is in the forefront of the battle against light pollution and you can find out more about the issues and solutions here.

But what about using a telescope? Clearly you can see fainter stars, but how much fainter?

There are a number of different formulae that have been proposed over the years. The answer will depend on the sky conditions, the observer’s eyesight and how efficient the telescope is at transmitting light. The equation below is a reasonable approximation for people with good eyesight, under a very dark clear sky and using a telescope with optics in good, clean condition. This is applicable to all types of telescope providing you use sufficient magnification so the exit pupil is smaller than the eye’s pupil and all the light leaving the eyepiece enters the eye.

m = 3.6 + 5log(D)
Where: m=faintest magnitude
D= diameter of telescope in millimetres.

From this we can calculate some examples for different size telescopes as in the table below.

Diameter
in mm
Diameter
in inches
Faintest
magnitude
75 3 13.0
100 4 13.6
125 5 14.1
150 6 14.5
200 8 15.1
250 10 15.6
300 12 16.0
350 14 16.3

Note that this limit is dependent on sky conditions and applies to point objects such as stars.

Limits for binoculars are more problematic, while using both eyes can give a small boost there are a number of factors which conspire to reduce the limiting magnitude. These include:

• Potential absorption and reflection of light by the binoculars internal prisms.
• Binoculars are often hand held and this will introduce some movement which will reduce the faintest magnitude available. This can be remedied at a price by using image stabilised binoculars or by mounting them on a tripod or parallelogram type mount.
• With binoculars usually having a low magnification there is the risk of the exit pupil being larger than the observer’s eye pupil and not all the light gathered by the binoculars actually entering the eye.

In theory, 10×50 binoculars should reach down to magnitude 12.1 but in practice this is seldom the case.

##### Surface brightness

All the magnitudes given so far are for objects that appear as point sources such as stars and planets (which appear as points even though they are actually small discs).

However there are also lots of objects in the sky that are not points of light but are extended or spread out over an area of sky. Typically these will be deep sky objects – star clusters, nebulae and galaxies although the same also applies to comets.

How is the magnitude scale applied to objects such as these? In this case the magnitude is based on the total light emitted across the whole object. This can mean that the quoted magnitude is not necessarily a reliable guide to how easy an object is to see. As an example take the spiral galaxy M33 in the constellation of Triangulum. This has a magnitude of 5.7 so should be readily visible to the naked eye under good conditions. Unfortunately, M33 is quite a large object and its light is spread out over an area of around four full moons making it a tricky object even in binoculars.

##### Absolute magnitudes

We now have a way of recording the apparent brightness of the stars but why do they differ in brightness? There are two reasons for this. Firstly they may be at differing distances.  If two stars are really the same brightness but one is further away than the other then clearly the more distant will appear fainter. The second reason is that some stars are truly more luminous than others.

These two factors mix up the situation and make it difficult to distinguish the truth about a star’s real brightness.

Clearly the apparent magnitude tells us nothing about the true brightness of a star, for this we have another measure, the absolute magnitude. This takes a star’s apparent magnitude and removes the effects of distance by recalculating how bright it would appear at a standard distance of 10 Parsecs which is equal to 32.6 light years.

The table below shows apparent and absolute magnitudes for a number of objects and also how bright they are in comparison to the Sun. Be aware that the distances to some of these stars are not known with total certainty and this can result in discordant values for absolute magnitude and relative brightness across different sources.

Object Apparent
Magnitude
Absolute
Magnitude
Brightness
relative to
the Sun
Sun -27 4.8 1.0x
Sirius – brightest star visible from Earth -1.44 1.4 24x
Vega – brightest star in Lyra 0.03 0.6 49x
Rigel – brightest star in Orion 0.18 -7.8 113,000x
Deneb – brightest star in Cygnus 1.25 -8.4 196,000x
Polaris – the Pole Star 1.97 -3.6 2,400x
Proxima Centauri – closest star to Earth 11.13 15.6 0.00005x

From this it is clear that the Sun is far less luminous than some of the stars we see in the night sky and there are those like Deneb which are true celestial searchlights. Then again there some very feeble lights that make our Sun seem a searchlight in its own right.

##### Different colours, different magnitudes

So far all the magnitudes we have discussed have been visual magnitudes, ones that tell us how bright a star appears as seen by the human eye with or without optical aid.

However, by using filters it is possible to isolate selected colours or wavelength bands within the star’s light and measure the magnitude for just that colour. When we do this we often find that a star’s magnitude is different in different colours.

A standard set of filters with carefully defined wavelength bands has been created and is known as the UBVRI system. The filters are as follows:

Filter Waveband Wavelengths
transmitted (nm)
U Ultra-violet 320-400
B Blue 400-500
V Visual 500-700
R Red 550-800
I Infra-red 700-900

The ‘V’ magnitude equates closely to that seen with the human eye and is what is used in the table of first magnitude stars shown earlier. From time to time you will see magnitudes quoted in the other bands as well, particularly in photometry – the precise measurement of stellar magnitudes.

The fact that stars have different brightness in different colours can tell us something about their makeup. A future tutorial will discuss how stars emit a whole spectrum of light, and by looking more closely at these spectra we can learn much about their magnitudes, likely age, masses and where they fit in to our understanding of the stars lives.

##### In conclusion

We have seen how a star’s brightness is noted to give its apparent magnitude and if we know its distance we can also calculate its absolute magnitude. From this we can get meaningful comparisons of the differences in the true brightness of the stars in the sky.

Remember that as the stars get fainter the magnitudes gets larger and that for an observer under a dark sky the naked eye limit is somewhere around sixth magnitude.

However not all stars are constant in their brightness. Many stars are variable and have magnitudes that change over periods from minutes to years. Paul Abel gives a good introduction to variable stars here. Variable star observing is a popular amateur pastime and one that can produce worthwhile scientific results. If this appeals to you why not contact the Variable Star Section of the BAA? They will be pleased to help and can provide guidance, variable star charts for binoculars and telescopes and even a mentoring scheme to help get you started.