1. Activity 2: Magnitudes and Colours of Stars
Swinburne Online Education Exploring Stars and the Milky Way
Module 4:
Vital Statistics of Stars
Module :
Activity 2: Magnitudes and Colours of Stars
© Swinburne University of Technology

2. Summary
In this Activity, we will investigate the magnitude and colour of stars. In particular, we will discuss:
the meaning of “colour” when applied to stars;
the meaning of magnitude;
the difference between apparent magnitude and absolute magnitude;
luminosity; and
the relationship between a star’s brightness and its size.

3. Stars: what we can measure
It would be lovely to be able to measure everything about stars directly...
… but until humans develop interstellar travel, the measurements we can make from observatories on Earth are limited.
Wow! What an interesting star ...
Scales too small, won’t reach ...
Meter too small, won’t reach ...
Thermometer won’t reach ...
Too far, picture too fuzzy ...
Too far, picture too fuzzy ...
Things to measure: temperature mass charge distance electromagnetic spectrum velocity relative to Earth age diameter energy output etc. etc. ? ? ? ? ?

4. What we can measure
All we can do is analyse the light that reaches the Earth from the star, and obtain data on:
the energy spectrum of the light emitted;
the energies missing from that spectrum;
Tell me about light
the intensity of the light emitted;
intensity
the distance of the star from Earth (using Cepheid variables or parallax for example).
However, as you will see, this is quite powerful data and we can do a surprising amount of science with it.
wavelength

5. Colour of Stars
The colour of a star depends on the strongest colour in its spectrum.
As you will learn in the next Module, the spectra of stars share some basic features, and there is a common “hill” shape, with a maximum.
Flux
wavelength
One with this spectrum will look yellow
A star with this spectrum will look blue
And one with this spectrum will look red
Visible light
Ultra-violet
Infra-red

6. Finding temperature from spectra
The position of the maximum in the spectrum from a star can indicate its surface temperature.
The hotter the star, the more light it emits at the blue, short wavelength end of the spectrum.
Ultraviolet
X-ray
gamma ray
Flux
Visible light
hot star
medium star
cool star
Infra-red
wavelength

7. Naos Rigel Sirius Canopus Arcturus Aldebaran Antares Betelgeuse
The colour of stars
This means that a star that looks blue is likely to be a very hot star, while a reddish star is (relatively speaking) cooler.
Our own Sun is a yellow star.
Flux
wavelength
Naos Rigel Sirius Canopus
hot star K
medium star K
Sun Capella
Arcturus Aldebaran Antares Betelgeuse
cool star K

8. Blue Stragglers
The colours of stars in these pictures are artificially enhanced, but still indicate the variation in colour within one star cluster.
The picture on the left was taken from the ground. The small rectangle marks the area detailed in the photo on the right, taken by the Hubble Space Telescope. The circles in the Hubble photo indicate “blue stragglers”: young, hot stars found in clusters of much older stars.

9. From spectrum to magnitude
The spectra of stars will be studied in more detail in the next Module “Colours and Spectral Types”.
Our next topic in this Activity is magnitude. We will have a look at:
how the brightness of stars is perceived;
how it is adjusted to compensate for our limited vision;
how it is adjusted for distance from the source; and
the definition of the various terms used in the process.

10. What is magnitude?
In astronomy, the magnitude of a star is a measure of how brightly it shines compared to other stars.
The human eye is good at telling the difference between stars that are roughly twice as bright as each other, but not much better than that.
Magnitude!
Magnitude?

11. Magnitude and History I II
Hipparchus in the 2nd century BC classified about a thousand stars into six brightness groups called “magnitudes”.
III
The first magnitude stars were the brightest.
They are about 100 times as bright as the sixth magnitude stars, which were the faintest stars that Hipparchus could see.
Each magnitude was about twice as bright as the next magnitude. IV V VI

12. Magnitude and Commonsense
Unfortunately, this has left us with a scale which runs counter to common sense in many ways, but its usage has a long history.
The way the magnitude scale is defined means that:
the brighter the star, the lower the magnitude
some stars (such as Sirius and the Sun) actually have a magnitude that is a negative number!
Sun
-26.5
Sirius
-1.5
Faintest detected
28.5
Aldebaran 1 10 20 30
-10
-20
-30
Naked eye
limit
~6.5
Binocular
limit ~9
Hubble
Space Telescope

13. I don’t know, but I can tell you how bright it looks ...
A very human scale
Although the magnitude scale seems strange at first, it is actually quite intuitive.
Hipparchus, who developed the magnitude scale, had little instrumentation and relied far more heavily on human senses than we do today.
The magnitude scale, along with other measures for things such as loudness and pitch of sound, is based not on a linear scale but rather on the way the human nervous system operates.
How bright is that star?
I don’t know, but I can tell you how bright it looks ...

14. Two ways of measuring Magnitude is measured on a logarithmic scale.
This means that increases in magnitude involve a multiplication, rather than a simple addition. Consider the following:
+ 1
+ 1
+ 1
0 km
1 km
2 km
3 km
4 km
5 km
Four kilometres is four times further than one kilometre, and twice as far as 2 kilometres. Each ‘one kilometre step’ involves the addition of an extra kilometre, so we say that...
Distance is measured on a linear scale.

15. So a sound 9 dB “louder” actually has 8 times the sound energy!
A logarithmic scale
So a sound 9 dB “louder” actually has 8 times the sound energy!
Our senses often don’t work in this way however.
What seems to a human observer to be a simple increase in loudness or pitch is actually not an addition but a multiplication.
3 dB “louder”
Twice the
intensity
3 dB “louder”
Twice the
intensity
In this example, each 3 decibel increase corresponds to a doubling in the intensity of sound.
3 dB “louder”
Twice the
intensity
Magnitude, like loudness, is measured on a logarithmic scale.

16. Our perception of the brightness of stars works in a similar fashion, and so the magnitude scale is a little odd at first.
x 100 I II
III IV V VI
x 2.512
x 2.512
x 2.512
x 2.512
x 2.512
It turns out that each step in magnitude corresponds to an increase in brightness by a factor of
Five even steps (from magnitude 1 to magnitude 6) give a factor of 100 in brightness.

17. Magnitude depends on where you are
The magnitude or brightness of a star depends on the distance between the star and the observer. The further away you are, the less bright the star will appear to be, according to the inverse square law.
The brightness we see from Earth is called the apparent magnitude.
Rubbish! It only looks about magnitude 6 from here!
That’s a fairly bright star: magnitude 2, I’d say
What’s the inverse square law?
We’d better call it the apparent magnitude, then

18. long wavelength low frequency short wavelength high frequency
Infra-red
long wavelength low frequency
Our Limited Vision
Human perception affects not only how we see brightness but what “colour” we think a star is.
Stars radiate at all kinds of wavelengths, from the ultraviolet to the infra-red and radio waves, and beyond.
However since magnitude was (and still is) defined by how stars appear to human vision, we may as well go the whole way and customise the definition a bit further to suit ourselves.
We can’t see in the infra-red, nor can we see in the ultraviolet, so why bother with them?
Ultraviolet
short wavelength high frequency

19. Apparent Visual Magnitude, mv
“v” is for “visual”
If we restrict our definition of magnitude to the “visual” wavelengths ( from roughly to angstroms, or 400 to 700 nanometers) then we have a pleasant and friendly measure by which we can classify and discuss stars and other objects.
The brightness of a star
apparent
visual
magnitude
… as seen from Earth
… by humans!

20. Stars radiate entire spectrum
For Earthers only
Venusians might might judge magnitude by these wavelengths
This definition of Apparent Visual Magnitude may not impress a Martian, but it certainly makes life easier for a life form which evolved on a planet where the atmosphere lets in lots of light in the nm range, and so that’s what our eyes can perceive.
Earth people judge magnitude by these wavelengths
Martians might judge magnitude by these wavelengths

21. Absolute Visual Magnitude, Mv
It’s all very well to say that a star appears dim in the visual wavelengths, but is it really dim? That will depend on its distance from you.
Astronomers use another measure, Absolute Visual Magnitude, to compensate for the fact that stars are at different distances from Earth. The Mv of a star is the magnitude that it would have if it were placed at 10 parsecs from Earth.
The apparent visual magnitude is 3
Real distance
Earth
10 parsec
star
… but the absolute visual magnitude is 0.8!

22. Some sample values of Mv
The apparent (mv) and absolute (Mv) visual magnitudes of some stars are given below, along with their distances (d) from our Solar System in light years.
Star mv Mv d(lyr)
Sun x10-5
Alpha Centauri A
Canopus
Rigel
Deneb

23. Some things to note
It will take some time for you to get used to magnitude measurements. For now, you should note that:
A large negative value means a very bright star, and
A large positive value means a very dim star.
Consider our own Sun…
The Sun looks extremely bright at the Earth’s surface ...
mv: measured at Earth
Mv: measured at 10 pc
Star mv Mv d
Sun (nil)
Alpha Centauri A
Canopus
Rigel
Deneb
but would look very dim at a distance of 10 pc

24. I think I will remember it this way:
What does that mean?
The faintest object we can comfortably perceive with the naked eye would have an apparent visual magnitude mv of +6.5 or so.
An object that was “pretty bright” would have an mv of about 0.
The Sun has an mv of – It is by far the brightest object in the sky.
Something with a plus sign is quite safe to look at, while a minus sign can mean danger!
I think I will remember it this way:

25. Twinkle, Twinkle, Little Star
Deneb: 490 pc away mv=1.26
Twinkle, Twinkle, Little Star
Consider the star Deneb.
Its apparent visual magnitude is 1.26, which means that it is a fairly bright star when seen from Earth.
If Deneb was 10 pc away Mv=-7.1
But its distance from Earth is about 490 pc, which means that in fact it shines very bright.
When the distance is taken into account, the result is an absolute visual magnitude of –7.1.
That makes Deneb one of the brightest objects around.

26. Star radiates heaps of energy in all directions
Luminosity
Luminosity is the amount of energy a star radiates in one second, and is often quoted relative to the luminosity of the Sun.
It would be nice to be able to simply measure the luminosity of a star directly, as this would help us to classify it and describe it (e.g. as incredibly luminous, low-luminosity etc).
… but only a tiny bit reaches Earth
However all we can measure is the tiny fraction of the radiation that reaches Earth. We can calculate the luminosity from this, but there are a few steps to fill in first.

27. Start with Magnitude mv + distance
1. Start with the measurement of apparent visual magnitude, mv.
This measurement can be made using modern photographic equipment (basically, a light meter!).
2. The distance to the star is also measured, using parallax. Mv
3. When both mv and the distance are known, the absolute visual magnitude Mv is calculated.

28. + non-visible radiation Absolute bolometric magnitudemv
+ distance Mv
Bolometric Magnitude
4. However the light considered in a measurement of mv (and then a calculation of Mv) is in the visual wavelengths only, so a correction must be made to include the wavelengths outside the visual range.
That way we can estimate the total energy radiated by the star at all wavelengths (not just the visual).
The result of the adjustment is called the absolute bolometric magnitude.
+ non-visible radiation
Absolute bolometric magnitude

29. + non-visible radiation Abs. bolometric magnitude = -0.3
mv = -0.06
An example: Arcturus
+ distance 11 pc
The apparent visual magnitude (as seen from Earth) of the star Arcturus is
Taking into account its distance of 11 pc gives an absolute visual magnitude of -0.3.
The adjustment to include non-visible wavelengths doesn’t make much difference in this case: the absolute bolometric magnitude is still about -0.3.
However if a star is very red or very blue the correction can be quite large.
Mv = -0.3
+ non-visible radiation
Abs. bolometric magnitude = -0.3

30. And finally, luminosity
abm of Acturus = -0.3
5. The absolute bolometric magnitude (abm) of the star is then compared to the absolute bolometric magnitude of the Sun (+4.7).
abm of Sun = +4.7
Difference = 5.0
Every difference of 1 in magnitude means a factor of in the luminosity of the star; a difference of 5 means a factor of 100.
… by definition, this means a factor of 100 in luminosity
luminosity of Sun = 4 x 1026 Watts
The luminosity of the Sun is known, so the luminosity of the star can be calculated.
so luminosity of Acturus = 4 x 1028 Watts

31. The Size of Stars
Let’s leave brightness for now, and start thinking about stellar size: another important property for classifying stars.
It is almost impossible to actually see a star through a telescope and measure its physical diameter. We can do this with objects within the Solar System, but the stars are simply too far away to appear as more than blurry dots.
How then, can astronomers confidently state that one star has a diameter a hundred times that of the Sun, while another has a diameter one-half that of the Sun?
Star* diameter** Frisbee Klaxon Microm
* not real stars ** relative to Sun
Well there are some clever observational tricks using pairs of telescopes known as interferometers, but there is often another easier indirect way ...

32. Luminosity Again temperature radius luminosity
… defines how much energy is given off per
square metre.
The luminosity of a star depends mostly on its temperature and its radius.
There is a simple physical law which determines how much radiation a black body emits. According to this law, luminosity, L, is related to temperature, T, and radius, R, by:
radius
… will determine the surface area of the star
 = …  = 5.98 x m-2 K-4 is the Stefan-Boltzmann constant.
luminosity
… the energy output of the star (per second)

33. How Hot, compared to the Sun?
We can measure the temperature of a star by looking at its spectrum. This will be studied more in the next Activity.
The hotter the star is, the bluer its light will be.
Comparing the spectra of stars lets us compare their temperatures.
Apparent
Brightness
Star’s spectrum
The spectra on the right show that the star is a lot hotter than our Sun, and would look blue to humans.
Sun’s spectrum
Wavelength

34. How Bright, compared to the Sun?
We can measure the luminosity of a star (by measuring its apparent visual magnitude and working back through the steps shown earlier in this Activity) until we can compare its luminosity to that of the Sun.
Apparent visual magnitude, mv
Compare to the Sun
Absolute visual magnitude, Mv
Absolute bolometric magnitude
Calculate luminosity
Distance from Earth

35. How Big?
So, if we know both the luminosity and the temperature, we can work out relative sizes of stars!
Here is an example about the reddish “star” Antares, which is actually a binary system (two stars orbiting each other) made up of
Antares A - reddish, surface temp.  3,000oK
Antares B - bluish-white, surface temp.  15,000oK
- but the combination looks reddish, because the measured light intensity that we pick up on Earth from Antares A  40 x that from Antares B,
and as they are close together, that means that Antares A must be approx. 40 times as luminous as Antares B.

36. How Big?
Surrounded by a nebula of expelled gas, Antares A is the brightest star in the constellation of Scorpio, and one of the brightest in the night sky.

37. Answer. It depends on their relative sizes.
Question. If Antares A is much cooler (3,000oK) than Antares B (15,000oK), how can it be so much more luminous than Antares B?
Answer. It depends on their relative sizes.
This is because the amount of energy radiated by each square metre of star’s surface does depend strongly on temperature ...

38. In fact, as we will see in the next Module, stars behave very like practical examples of black bodies - theoretical objects with properties that have been determined by classical physicists. (We will leave the tricky question of how someone could describe a star as a black body to the next Module!)
This is very useful, because classical physicists in the 1800s worked out lots of laws which apply to black bodies. One in particular, the Stefan-Boltzmann Law, is very important for the study of stars.

39. Stefan-Boltzmann Law
The Stefan-Boltzmann Law says that if an object radiates like a black body, then the amount of energy, E, it gives out per second (its luminosity) is related to its temperature T by
… for each square metre of its surface.
Note the emphasis on each square metre of its surface - if two stars are about the same size, the hotter one will definitely be by far the most luminous,
… but a huge cool star can still radiate more than a small hot star, because of all its extra square metres of surface.
 = 5.98 x m-2 K-4 is the Stefan-Boltzmann constant we saw earlier.

40.  It turns out that Antares A manages to be so much more luminous (x 40) than Antares B, even though it is a factor of five cooler, because it is 160 times bigger than Antares B, and about 700 times the diameter of our Sun.
(in diameter) .
Antares B
(Antares A is a red supergiant star - more about these in the Module on Stellar Old Age.)
Antares A

41. Summary
This Activity has shown you how some of the simple measurements that we can make in astronomy can lead to really interesting and exciting facts about stars and other distant objects.
In spite of the fact that we are stuck on Earth, our instruments can measure the position of a star as the year passes, the brightness of a star, and the spectrum of light from a star.
These allow us to calculate things that we can’t possibly measure directly, such as the distance to the star, the luminosity of the star, the temperature and the size of the star.
In the next Module, we will look more closely at stellar spectra.

42. Image Credits
The Earth’s Moon:
AAO: Antares © David Malin (used with permission)

43. Hit the Esc key (escape) to return to the Module 4 Home Page

45. Inverse square law
If something is being emitted with equal intensity in all directions from a point source, it will obey the
“Inverse Square Law”.
Point source of light
Closer in, the intensity of light is high as the light is only spread over a small area
Further out, the intensity of light is low as the light is spread over a larger area

46. Imagine that a star is emitting light equally in all directions.
At planet Alpha, the light is observed as being fairly intense, as it is being shared over a small area:
small radius, therefore
small area, therefore
high light intensity.
Star
At planet Beta, the light is being shared over a larger area and so the intensity of the light is far less:
large radius, therefore
large area, therefore
low light intensity.

47. Return to Activity

48. Light
For thousands of years people tried to come up with a description of light which explained the many surprising things that light can do, including:
reflection;
refraction;
dispersion; and
interference, especially diffraction.
Since astronomers learn about other places in the Universe by studying the light we receive from them, you will need to know a bit about light during your study of astronomy.
That star has got light and dark patches around it!
No it hasn’t. The star isn’t like that; it’s our telescope’s fault

49. An Example of Diffraction
This photo shows “spikes” and rings caused by diffraction through the telescope instrumentation:
These rings and spikes are diffraction effects

50. It is neither, but it’s like both
What is light?
Although reflection can be explained easily enough if light is made up of tiny particles, it is very hard to explain phenomena like diffraction unless light is a wave.
Is it a wave?
A wave model has its own problems, however. For instance, if light is a wave, and waves require a medium such as air or water to carry them, how does light travel through empty space?
Is it a particle?
It is neither, but it’s like both
Physicists now believe that light is neither a wave nor a particle, but its behaviour sometimes resembles a wave and sometimes a particle.

51. Waves
It is convenient to treat light as a wave when discussing colour, and this is a property of prime importance to astronomers (particularly when examining the colours of stars, as we are doing in this Activity).
Waves are described in physics by a few standard dimensions.
Amplitude A = height of wave above “rest position”
Wavelength  = length of one cycle
Velocity v = speed of wave  A
Frequency f = how often the wave passes: longer wavelength means lower frequency

52. Frequency and energy Waves in general Light
Frequency is very important in physics and in astronomy, where we are very often interested in such things as energy and temperature.
This is because energy, E, is related to the frequency of light by the formula:
= x Js
When writing about light, people often use the Greek symbol  (pronounced “noo”) for frequency, and c for the speed of light.
In astronomy, you will often see the symbols  and c for frequency and speed.
Waves in general
Light

53. Electromagnetic radiation
Light is just one type out of many types of “electromagnetic radiation” (EMR).
Electrons accelerate and decelerate
Electrons drop to lower energy levels
EMR is produced when electrons decelerate and lose energy (e.g. in a radio transmitter)
releasing energy in the form of EMR
releasing energy in the form of EMR
or drop from a high energy level in an atom to a lower one (e.g. in the chromosphere of a star), and lose energy.

54. Observing EMR
When EMR is absorbed or detected (e.g. by a leaf, an eye, a telescope or photographic film) the reverse happens.
EMR is absorbed by electrons
EMR is absorbed by electrons
allowing a reaction to take place
and is turned into an electrical signal
The energy of the EMR is absorbed by electrons and converted to electrical energy (e.g. in a solar panel)
or it causes an electron to jump to a higher energy level, allowing a chemical reaction to take place (e.g. in the human eye).

55. Colour of electromagnetic radiation
The human eye interprets difference in frequency as “colour”, and calls the range of frequencies that we can see “visible light”.
450 nm
700 nm
wavelength
ultraviolet
infra-red
frequency
6 x 1014 Hz
There are an infinite number of possible frequencies (and wavelengths) for light, but humans can see only a very small band of them between the ultraviolet and the infra-red.

56. Return to Activity