Transcript Magnitudes and Colours of Stars - Lincoln

Swinburne Online Education Exploring Stars and the Milky Way
Module
: 4:
Module
Vital Statistics of Stars
Activity 2:
Magnitudes and
Colours of Stars
© Swinburne University of Technology
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.
Stars: what we can measure
It would be lovely to be able to measure everything
about stars directly...
Scales
Meter
Wow!
Thermometer
Too
too
What
too
far,
small,
small,
an
picture
interesting
won’ttoo
reach
fuzzy
star
.........
?
?
?
? ?
… but until humans
develop interstellar
travel, the
measurements
we can make
from observatories
on Earth
are limited.
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 intensity of the light emitted;
intensity
Tell me
about
light
• 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
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
One with this
spectrum will
look yellow
A star with this
spectrum will
look blue
Ultra-violet
And one with
this spectrum
will look red
Visible light
Infra-red
wavelength
Finding temperature from spectra
Flux
The position of the maximum in the spectrum from
a star can indicate its surface temperature.
The hotter the star, the more light
Ultraviolet
X-ray
it emits at the blue, short wavelength
gamma ray
end of the spectrum.
Visible light
hot star
medium star
cool star
Infra-red
wavelength
The colour of stars
Flux
hot star
11 000 - 30 000 K
Naos
Rigel
Sirius
Canopus
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
medium star
a yellow star.
5 000 - 11 000 K
Sun
Capella
Arcturus
Aldebaran
Antares
Betelgeuse
cool star
3 500 - 5 000 K
wavelength
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.
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.
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!
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
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
-30
Sirius
-1.5
-20
-10
Aldebaran
1
0
Naked eye
limit
~6.5
10
Binocular
limit
~9
Faintest detected
28.5
20
30
Hubble
Space
Telescope
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 ...
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
0 km
+1
1 km
+1
2 km
+1
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.
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.
In this example, each 3 decibel
increase corresponds to a doubling
in the intensity of sound.
So a sound 9 dB “louder”
actually has 8 times the
sound energy!
3 dB
“louder”
Twice the
intensity
3 dB
“louder”
Twice the
intensity
3 dB
“louder”
Twice the
intensity
Magnitude, like loudness, is measured on a logarithmic scale.
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
x 2.512
x 2.512
III
x 2.512
IV
x 2.512
V
VI
x 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.
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 the
apparent magnitude, then
What’s
the inverse
square law?
Our Limited Vision
Infra-red
long wavelength
low frequency
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
Apparent Visual Magnitude, mv
“v” is for “visual”
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.
The brightness of a star
apparent
… as seen from Earth
visual magnitude
… by humans!
Stars radiate
entire spectrum
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.
Venusians might
might judge
magnitude by
these wavelengths
Earth people
judge magnitude
by these wavelengths
Martians might
judge magnitude
by these
wavelengths
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
10 parsec
Earth
star
… but the absolute visual
magnitude is 0.8!
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
Sun
Alpha Centauri A
Canopus
Rigel
Deneb
mv
-26.8
-0.27
-0.72
0.14
1.26
Mv
4.83
4.4
-3.1
-7.1
-7.1
d(lyr)
1.7x10-5
4.3
98
900
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.
Consider our own Sun…
mv: measured
atmEarth
Star
Sun
Alpha Centauri A
Canopus
Rigel
Deneb
v
-26.8
-0.27
-0.72
0.14
1.26
Mv: measured at
M10 pc
v
4.83
4.4
-3.1
-7.1
-7.1
The Sun looks
extremely bright
d Earth’s
at the
(nil) ...
surface
but 4.3
would look
98dim at a
very
900
distance
of
1600
10 pc
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 will
remember it
this way:
Something with a plus sign
is quite safe to look at,
while a minus sign
can mean danger!
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.
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.
Deneb:
490 pc away
mv=1.26
If Deneb was
10 pc away
Mv=-7.1
Luminosity
Star radiates
heaps of energy
in all directions
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.
… but only
a tiny bit
reaches
Earth
Start with Magnitude
mv
+
1. Start with the measurement of apparent visual distance
magnitude, mv.
This measurement can be made using modern
photographic equipment (basically, a light meter!).
Mv
2. The distance to the star is also measured, using
parallax.
3. When both mv and the distance are known, the
absolute visual magnitude Mv is calculated.
mv
Bolometric Magnitude
+
distance
4. However the light considered in a
measurement of mv (and then a calculation
of Mv) is in the visual wavelengths only, so a
Mv
correction must be made to include the
+
wavelengths outside the visual range.
That way we can estimate the total energy non-visible
radiation
radiated by the star at all wavelengths (not
just the visual).
The result of the adjustment is called
Absolute
the absolute bolometric magnitude.
bolometric
magnitude
mv = -0.06
An example: Arcturus
+
The apparent visual magnitude (as seen distance 11 pc
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.
Mv = -0.3
The adjustment to include non-visible
+
wavelengths doesn’t make much difference
non-visible
in this case: the absolute bolometric
radiation
magnitude is still about -0.3.
However if a star is very red or very blue the
correction can be quite large.
Abs. bolometric
magnitude = -0.3
And finally, luminosity
abm of Acturus
= -0.3
5. The absolute bolometric
abm of Sun
magnitude (abm) of the star is then
= +4.7
compared to the absolute bolometric
Difference = 5.0
magnitude of the Sun (+4.7).
… by definition, this
Every difference of 1 in magnitude
means a factor of 100
means a factor of 2.512 in the
in luminosity
luminosity of the star; a difference of
luminosity of Sun
5 means a factor of 100.
26 Watts
=
4
x
10
The luminosity of the Sun is known,
so the luminosity of the star can be
calculated.
so luminosity of
Acturus
= 4 x 1028 Watts
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?
Well there are some clever observational
tricks using pairs of telescopes known
as interferometers, but there is often another
easier indirect way ...
Luminosity Again
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:
 = 3.1415926…
 = 5.98 x 10-8 m-2 K-4 is the StefanBoltzmann constant.
temperature
… defines how much
energy is given off per
square metre.
radius
… will determine
the surface area of the
star
luminosity
… the energy output of
the star (per second)
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.
The spectra on the right
show that the star is a lot
hotter than our Sun, and
would look blue to humans.
Apparent
Brightness
Star’s spectrum
Sun’s spectrum
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.
Apparent
visual
magnitude,
mv
Distance
from Earth
Absolute
visual
magnitude,
Mv
Absolute
bolometric
magnitude
Compare
to the
Sun
Calculate
luminosity
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
Antares A - reddish,
(two stars orbiting
surface temp.  3,000oK
each other) made up of
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 expelled
gas, Antares A is
the brightest star
in the constellation
of Scorpio, and
one of the brightest
in 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-Boltzmann
Law, is very important for the study of stars.
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 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 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
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.
Image Credits
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
Hit the Esc key (escape)
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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
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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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
That star has
studying the light we receive
got light and dark
from them, you will need to
patches around it!
know a bit about light during
No it hasn’t.
your study of astronomy.
The star isn’t like that;
it’s our telescope’s fault
An Example of Diffraction
This photo shows “spikes” and rings caused by diffraction through
the telescope instrumentation:
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 Is it a wave?
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 particle?
It is neither,
but it’s
like both
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.
Wavelength 
= length of one cycle
Amplitude A
= height of wave
above “rest position”

A
Frequency f = how often the wave passes:
longer wavelength means lower frequency
Velocity v
= speed of wave
Frequency and energy
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:
= 6.626 x 10-34 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
Electromagnetic radiation
Electrons
accelerate and
decelerate
releasing
energy in the
form of EMR
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 drop
to lower energy
levels
releasing
energy in the
form of EMR
Observing EMR
EMR is
absorbed by
electrons
and is turned
into an electrical
signal
When 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
allowing a
reaction to take
place
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
ultraviolet
frequency
700 nm wavelength
infra-red
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.
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