Transcript Slide 1

Lecture 2
The distance scale
Apparent magnitudes
The magnitude system expresses fluxes in a given
waveband X, on a relative, logarithmic scale:
 f 

m X  mref  2.5 log 
f 
 ref 
 Note the negative sign means brighter objects have
lower magnitudes
 Scale is chosen so that a factor 100 in brightness
corresponds to 5 magnitudes (historical)
The magnitude scale
 f 

m X  mref  2.5 log 
f 
 ref 
One common system is to measure relative to Vega
 By definition, Vega has m=0 in all bands. Note this does not mean Vega is equally
bright at all wavelengths!
 Setting mref=0 in the equation above gives:
mX  2.5 log  f   2.5 log  fVega, X 
 2.5 log  f   m0, X
• Colour is defined as the relative flux between two different
wavebands, usually written as a difference in magnitudes
Apparent magnitudes
The faintest (deepest) telescope image
taken so far is the Hubble Ultra-Deep
Field. At m=29, this reaches more than
1 billion times fainter than what we can
see with the naked eye.
 f 

m X  mref  2.5 log 
f 
 ref 
Object
Apparent
mag
Sun
-26.5
Full moon
-12.5
Venus
-4.0
Jupiter
-3.0
Sirius
-1.4
Polaris
2.0
Eye limit
6.0
Pluto
15.0
Reasonable telescope limit (8-m
telescope, 4 hour integration)
28
Deepest image ever taken
(Hubble UDF)
29
10( 296) / 2.5  1046 / 5  109
Imagine a hypothetical source which has a constant flux of
10 Jy at all frequencies. What is its magnitude in the U
band? In the V and K bands?
Band Central
Bandwidth Flux of Vega
name Wavelength (mm)
(Jy)
(mm)
U
B
0.37
0.45
0.066
0.094
1780
4000
V
0.55
0.088
3600
R
I
0.66
0.81
0.14
0.15
3060
2420
J
H
K
1.25
1.65
2.20
0.21
0.31
0.39
1570
1020
636
mX  2.5 log  f   2.5 log  fVega, X 
 2.5 log  f   m0, X
What is the B-V colour of a source that has a flux
proportional to l-4?
Band Central
Bandwidth Flux of Vega
name Wavelength (mm)
(Jy)
(mm)
U
B
0.37
0.45
0.066
0.094
1780
4000
V
0.55
0.088
3600
R
I
0.66
0.81
0.14
0.15
3060
2420
J
H
K
1.25
1.65
2.20
0.21
0.31
0.39
1570
1020
636
mX  2.5 log  f   2.5 log  fVega, X 
 2.5 log  f   m0, X
Absolute magnitudes
It is also useful to have a measurement of intrinsic brightness that is
independent of distance
F
L
4r 2
Absolute Magnitude (M) is therefore defined to be the magnitude a star
would have if it were at an arbitrary distance D0=10pc:
 10 pc 

m  M  5 log 
 Dstar 
D 
 5 log  star   5
 pc 
(note the zeropoints have cancelled)
The value of m-M is known as the distance modulus.
Example
Calculate the apparent magnitude of the Sun (absolute
magnitude M=4.76) at a distance of 1 Mpc (106 pc)
 Dstar 
m  M  5 log 
5

 pc 
• Recall that the deepest
exposures taken reach m=29
• The nearest large galaxy to us is
Andromeda (M31), at a distance of
about 1 Mpc
 Detecting stars like our Sun in
other galaxies is therefore very
difficult (generally impossible at
the moment).
The colour-magnitude diagram
Precise parallax
measurements allow us to
plot a colour-magnitude
diagram for nearby stars.
 The HertzsprungRussel (1914) diagram
proved to be the key
that unlocked the
secrets of stellar
evolution
 Colour is independent
of distance, since it is
a ratio of fluxes:
f red 4r 2 Lred Lred


f blue 4r 2 Lblue Lblue
 Absolute magnitude
(y-axis) requires
measurement of flux
and distance
Types of stars
Intrinsically faint stars are more
common than luminous stars
Main sequence fitting
NGC2437
Stellar clusters:
 Consist of many, densely packed stars
 For distant clusters, it is a very good approximation that all the
constituent stars are the same distance from us.
 Typical clusters have sizes ~1 pc; so for clusters >10 pc away this
assumption introduces a 10% error.
 Therefore, we can plot a colour-magnitude diagram using only the
apparent magnitude on the y-axis, and recognizable structure appears.
Main sequence fitting
Nearby stars (parallax)
distant cluster (apparent magnitudes)
We can take advantage of the structure in the HR diagram to determine
distances to stellar clusters
 Colour is independent of distance, so the vertical offset of the main
sequence gives you the distance modulus m-M
Main sequence fitting
Example: NGC2437:
At a colour of B-V=1.0 mag, the main sequence absolute magnitude is 6.8.
In NGC2437, at the same colour, V=17.5. Thus the distance modulus is:
DM  V  M V
 10.7
 5 log d  5
This gives a distance of 1.4 kpc to NGC2437, reasonably close to the accepted distance of 1.8 kpc.
Break
Variable stars
The images above show the same star field at two different times. One
of the stars in the field has changed brightness relative to the other stars
– can you see which one?
Variable stars
The images above show the same star field at two different times. One
of the stars in the field has changed brightness relative to the other stars
– can you see which one?
Variable stars
•Many stars show fluctuations in
their brightness with time.
•These variations can be
characterized by their light curve –
a plot of their magnitude as a
function of time
Variable stars
Certain intrinsically variable stars show a
remarkably strong correlation between
their pulsation period and average
luminosity
Modern calibration of the
Cepheid P-L relation in the
Magellanic clouds, yields:
M I  2.96(log P 1)  4.9
Where the period P is measured in days, and the
magnitude is measured in the I band.
Instability strip
• Classical Cepheids are not the
only type of pulsating variable
star, however
• There is a narrow strip in the HR
diagram where many variable
stars lie
• Cepheids are the brightest
variable stars; however they are
also very rare
Cepheids
W Virginis
RR Lyrae
Pulsating white
dwarfs
RR Lyrae Stars
RR Lyrae stars (absolute magnitudes M=+0.6) are much
fainter than Cepheids; but have the advantage that they
almost all have the same luminosity and are more
common. They are easily identified by their much
shorter periods
Period (days)
Absolute Magnitude
Schematic
representation
Log (Period)
RR Lyrae variables
RR Lyrae stars have average absolute magnitudes
M=+0.6. How bright are these stars in Andromeda?
Summary: the distance ladder
1. Find parallax distances to the nearest stars
•
•
Dedicated satellites are now providing these precise
measurements for thousands of stars
Plot stellar absolute magnitudes as a function of colour
2. Measure fluxes and colours of stars in distant clusters
•
•
Compare with colour-magnitude diagram of nearby stars (step 1)
and use main-sequence fitting method to compute distances
Identify any variable stars in these clusters. Calibrate a periodluminosity relation for these variables
3. Measure the periods of bright variable stars in remote
parts of the Galaxy, and even in other galaxies
•
Use the period-luminosity relation from step 2 to determine the
distance
Note how an error in step 1 follows through all subsequent
steps!
Spectroscopy
In 1814, Joseph Fraunhofer catalogued
475 sharp, dark lines in the solar spectrum.
• Discovered but misinterpreted in
1804 by William Wollaston
• Spectrum was obtained by passing
sunlight through a prism
Example: the solar spectrum
What elements are present in the Sun?
Solar spectrum
Example: the solar spectrum
What elements are present in the Sun?
Balmer lines (Hydrogen)
Example: the solar spectrum
What elements are present in the Sun?
NaD
Example: the solar spectrum
What elements are present in the Sun?
Ca H+K
Example: the solar spectrum
So: the Sun is mostly calcium, iron and sodium?? No! Not quite that simple…
Solar spectrum
Stellar spectra
Increasing temperature
Stellar spectra show interesting trends as a function of temperature:
Spectral classification
Stars can be classified according to the relative strength of
their spectral features:
 There are seven main classes, in order of decreasing
temperature they are: O B A F G K M
 For alternative mneumonics to the traditional ‘O be a fine girl kiss
me’, see here
 Each class is subdivided more finely from 0-9. So a B2 star is
hotter than a B9 which is hotter than a A0
 Additional classes are R, N, S which are red, cool supergiant
stars with different chemical compositions
Characteristics of spectral classes
Spectral
Type
Colour
Temperature Main characteristics
(K)
Example
O
Blue-white
>25000
10 Lacertra
B
Blue-white
11000-25000
Strong HeII absorption (sometimes
emission); strong UV continuum
HeI absorption, weak Balmer lines
A
White
7500-11000
Strongest Balmer lines (A0)
Sirius
F
Yellow-white 6000-7500
CaII lines strengthen
Procyon
G
Yellow
5000-6000
Solar-type spectra
Sun
K
Orange
3500-5000
Strong metal lines
Arcturus
M
Red
<3500
Molecular lines (e.g. TiO)
Betelgeuse
Rigel
Luminosity
The HR diagram revisited
O
B
Spectral Class
A
F
G
K
M
Henry Norris’ original diagram, showing stellar luminosity as a
function of spectral class.
The original HR diagram
The main sequence is clearly visible
A modern colour-magnitude
diagram