Module :

Activity 2:

Magnitudes and Colours of Stars

Module 4: Vital Statistics of Stars

Swinburne Online Education Exploring Stars and the Milky Way

© Swinburne University of Technology

SummaryIn 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.

Stars: what we can measure

It would be lovely to be able to measure everything about stars directly...

Things to measure:

temperature

mass

charge

distance

electromagnetic spectrum

velocity relative to Earth

age

diameter

energy output

etc. etc.

? ?

?

? ?

… but until humans develop interstellar travel, the measurements

we can make from observatories

on Earth are limited.

Wow! What an interesting star ...

Thermometerwon’t reach ...

Scales too small,won’t reach ...

Meter too small,won’t reach ...

Too far,picture too fuzzy ...

Too far,picture too fuzzy ...

• the intensity of the light emitted;

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;

• 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.

intensity

wavelength

Tell me about light

Colour of StarsThe 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

And one with this spectrum will look red

And one with this spectrum will look red

One with this spectrum will look yellow

One with this spectrum will look yellow

A star with this spectrum will

look blue

A star with this spectrum will

look blue

Visible lightUltra-violet Infra-red

The position of the maximum in the spectrum from a star can indicate its surface temperature.

The hotter the star, the more lightit emits at the blue, short wavelength

end of the spectrum.

Finding temperature from spectra

Ultraviolet

X-ray

gamma ray

Ultraviolet

X-ray

gamma ray

Flux

wavelength

medium star

cool star

hot star

Visible lightVisible light

Infra-redInfra-red

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.

The colour of stars

Flux

wavelength

NaosRigelSirius

Canopus

NaosRigelSirius

Canopus

hot star11 000 - 30 000 K

hot star11 000 - 30 000 K medium star

5 000 - 11 000 Kmedium star

5 000 - 11 000 KSun

CapellaSun

Capella

ArcturusAldebaran

AntaresBetelgeuse

ArcturusAldebaran

AntaresBetelgeuse

cool star3 500 - 5 000 K

cool star3 500 - 5 000 K

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.

Blue Stragglers

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.

From spectrum to magnitude

Magnitude?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.

What is magnitude?

Magnitude and History

Hipparchus in the 2nd century BC classified about a thousand stars into six brightness groups called “magnitudes”.

I

II

III

IV

V

VI

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 twiceas bright as the next magnitude.

Magnitude and CommonsenseUnfortunately, 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!

0 10 20 30-10-20-30

Sun-26.5

Aldebaran1

Sirius-1.5

Faintest detected28.5

Naked eye limit ~6.5

Naked eye limit ~6.5

Binocularlimit~9

Binocularlimit~9

HubbleSpace

Telescope

HubbleSpace

Telescope

A very human scaleAlthough 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 brightit looks ...

Two ways of measuring

This means that increases in magnitude involve a multiplication, rather than a simple addition. Consider the following:

Distance is measured on a linear scale. Distance is measured on a linear scale.

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...

Magnitude is measured on a logarithmic scale.

+ 1+ 1 + 1+ 1 + 1+ 1+ 1+ 1

A logarithmic scale

In this example, each 3 decibel increase corresponds to a doubling in the intensity of sound.

3 dB “louder”

3 dB “louder”

3 dB “louder”

3 dB “louder”

3 dB “louder”

3 dB “louder”

Twice theintensity

Twice theintensity

Twice theintensity

Twice theintensity

Twice theintensity

Twice theintensity

So a sound 9 dB “louder” actually has 8 times the

sound energy!

So a sound 9 dB “louder” actually has 8 times the

sound energy!

Magnitude, like loudness, is measured on a logarithmic scale. Magnitude, like loudness, is measured on a logarithmic scale.

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.

Our perception of the brightness of stars works in a similar fashion, and so the magnitude scale is a little odd at first.

I II III IV V VI

x 100x 100

x 2.512x 2.512x 2.512x 2.512x 2.512x 2.512x 2.512x 2.512x 2.512x 2.512

It turns out that each step in magnitude corresponds to an increase in brightness by a factor of 2.512.

Five even steps (from magnitude 1 to magnitude 6) give a factor of 100 in brightness.

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.

What’s the inverse square law?

That’s a fairly bright star: magnitude 2, I’d say

Rubbish! It only looks about magnitude 6 from here!

We’d better call it theapparent magnitude, then

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.

Infra-redlong wavelengthlow frequency

Infra-redlong wavelengthlow frequency

Ultravioletshort wavelengthhigh frequency

Ultravioletshort wavelengthhigh frequency

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?

Apparent Visual Magnitude, mv

If we restrict our definition of magnitude to the “visual” wavelengths ( from roughly 4 000 to 7 000 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.

magnitudemagnitudeapparentapparent visualvisual

The brightness of a star

… as seen from Earth… by humans!

“v” is for “visual”

For Earthers only

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 400-700 nm range, and so that’s what our eyes can perceive.

Stars radiateentire spectrum

Stars radiateentire spectrum

Venusians mightmight judge

magnitude bythese wavelengths

Venusians mightmight judge

magnitude bythese wavelengths

Earth peoplejudge magnitude

by these wavelengths

Earth peoplejudge magnitude

by these wavelengths

Martians mightjudge magnitude

by thesewavelengths

Martians mightjudge magnitude

by thesewavelengths

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.

star

EarthReal distance

10 parsec

The apparent visualmagnitude is 3

… but the absolute visualmagnitude is 0.8!

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.

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 -26.8 4.83 1.7x10-5

Alpha Centauri A -0.27 4.4 4.3

Canopus -0.72 -3.1 98

Rigel 0.14 -7.1 900

Deneb 1.26 -7.1 1600

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.

Star mv Mv dSun -26.8 4.83 (nil)Alpha Centauri A -0.27 4.4 4.3Canopus -0.72 -3.1 98Rigel 0.14 -7.1 900Deneb 1.26 -7.1 1600

mv: measured at Earth

mv: measured at Earth

Mv: measured at 10 pc

Mv: measured at 10 pc

The Sun looks extremely bright

at the Earth’s surface ...

The Sun looks extremely bright

at the Earth’s surface ...

but would look very dim at a distance of

10 pc

but would look very dim at a distance of

10 pc

Consider our own Sun…

Something with a plus signis quite safe to look at,

while a minus signcan mean danger!

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 –26.8. It is by far the brightest object in the sky.

I think I willremember it

this way:

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.

Deneb:490 pc away

mv=1.26

If Deneb was10 pc away

Mv=-7.1

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.

But its distance from Earth is about 490 pc, which means that in fact it shines very bright.

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).

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.

Star radiates heaps of energyin all directions

Star radiates heaps of energyin all directions

… but only a tiny bit reaches

Earth

… but only a tiny bit reaches

Earth

Start with Magnitude

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.

mvmv

+distance

+distance

MvMv

3. When both mv and the distance are known, the absolute visual magnitude Mv is calculated.

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 calledthe absolute bolometric magnitude.

mvmv

+distance

+distance

MvMv

+non-visibleradiation

+non-visibleradiation

Absolute bolometric magnitude

Absolute bolometric magnitude

An example: Arcturus

The apparent visual magnitude (as seen from Earth) of the star Arcturus is -0.06.

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 = -0.06mv = -0.06

+distance 11 pc

+distance 11 pc

Mv = -0.3Mv = -0.3

+non-visibleradiation

+non-visibleradiation

Abs. bolometric magnitude = -0.3Abs. bolometric magnitude = -0.3

And finally, luminosity5. The absolute bolometric magnitude (abm) of the star is then compared to the absolute bolometric magnitude of the Sun (+4.7).

abm of Acturus= -0.3

abm of Acturus= -0.3

Difference = 5.0Difference = 5.0… by definition, this

means a factor of 100 in luminosity

… by definition, this means a factor of 100

in luminosity

abm of Sun= +4.7

abm of Sun= +4.7

so luminosity of Acturus

= 4 x 1028 Watts

so luminosity of Acturus

= 4 x 1028 Watts

luminosity of Sun= 4 x 1026 Watts

luminosity of Sun= 4 x 1026 Watts

Every difference of 1 in magnitude means a factor of 2.512 in the luminosity of the star; a difference of 5 means a factor of 100.

The luminosity of the Sun is known, so the luminosity of the star can be calculated.

The Size of StarsLet’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 4.8

Klaxon 100

Microm 0.5

* not real stars

** relative to Sun

Star* diameter**

Frisbee 4.8

Klaxon 100

Microm 0.5

* not real stars

** relative to Sun

Well there are some clever observational tricks using pairs of telescopes knownas interferometers, but there is often another easier indirect way ...

Luminosity AgainThe luminosity of a star depends mostly on its temperature and its radius.

luminosityluminosity

temperaturetemperature… defines how much

energy is given off persquare metre.

… defines how much energy is given off per

square metre.

… the energy output of the star (per second)

… the energy output of the star (per second)

radiusradius… will determine

the surface area of the star

… will determinethe surface area of the

star

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:

= 3.1415926… = 5.98 x 10-8 m-2 K-4 is the Stefan-Boltzmann constant.

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.

Star’s spectrumStar’s spectrum

Sun’s spectrumSun’s spectrum

The spectra on the right show that the star is a lot hotter than our Sun, and would look blue to humans.

ApparentBrightness

Wavelength

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.

Distance from EarthDistance

from Earth

Absolute visual

magnitude, Mv

Absolute visual

magnitude, Mv

Absolute bolometricmagnitude

Absolute bolometricmagnitude

Calculate luminosityCalculate

luminosity

Apparent visual

magnitude, mv

Apparent visual

magnitude, mv

Compare to the Sun

Compare to the Sun

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.

How Big?

Surrounded by a nebula of expelledgas, Antares A isthe brightest starin the constellation of Scorpio, and one of the brightestin the night sky.

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 ...

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-BoltzmannLaw, is very important for the study of stars.

Stefan-Boltzmann LawThe 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

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,

… for each square metre of its surface.

… 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 10-8 m-2 K-4 is the Stefan-Boltzmann constant we saw earlier.

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 timesbigger than Antares B, and about 700 times the diameter of our Sun.

(in diameter)

.

Antares B

Antares A

(Antares A is a redsupergiant star - more about these in the Module on Stellar Old Age.)

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.

Summary

The Earth’s Moon: http://nssdc.gsfc.nasa.gov/planetary/banner/moonfact.gif

AAO: Antares © David Malin (used with permission)http://antwrp.gsfc.nasa.gov/apod/image/9706/antaresneb_uks_big.jpg

Image Credits

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

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”.

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

Point source of light

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.

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.

Star

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LightFor 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 hasgot light and darkpatches around it!

No it hasn’t.The star isn’t like that;

it’s our telescope’s fault

An Example of DiffractionThis photo shows “spikes” and rings caused by diffraction through the telescope instrumentation:

These rings and spikes are diffraction

effects

These rings and spikes are diffraction

effects

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.

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?

Physicists now believe that light is neither a wave nor a particle, but its behaviour sometimes resembles a wave and sometimes

a particle.

Is it a wave?

Is it a particle?

It is neither,but it’s

like both

WavesIt 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.

Wavelength = length of one cycle

Wavelength = length of one cycle

Amplitude A= height of wave

above “rest position”

Amplitude A= height of wave

above “rest position”

Frequency f = how often the wave passes:longer wavelength means lower frequency

Frequency f = how often the wave passes:longer wavelength means lower frequency

A

Velocity v= speed of wave

Velocity v= speed of wave

Waves in generalWaves in general LightLight

Frequency and energyFrequency 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:

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.

= 6.626 x 10-34 Js

Electromagnetic radiation

Light is just one type out of many types of “electromagnetic radiation” (EMR).

EMR is produced when electrons decelerate and lose energy (e.g. in a radio transmitter)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.

Electrons accelerate and

deceleratereleasing

energy in the form of EMR

Electrons drop to lower energy

levelsreleasing

energy in the form of EMR

Observing EMRWhen EMR is absorbed or detected (e.g. by a leaf, an eye, a telescope or photographic film) the reverse happens.

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).

EMR is absorbed by electrons

and is turned into an electrical

signal

EMR is absorbed by

electronsallowing a

reaction to take place

Colour of electromagnetic radiationThe human eye interprets difference in frequency as “colour”, and calls the range of frequencies that we can see “visible light”.

wavelength

frequency

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.

450 nm 700 nm

ultraviolet infra-red

6 x 1014 Hz

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