Post on 14-Dec-2015
Lecture 2Lecture 2
The distance scale
Apparent magnitudesApparent magnitudes
The magnitude system expresses fluxes in a given waveband X, on a relative, logarithmic scale:
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)
refrefX f
fmm log5.2
The magnitude scaleThe magnitude scale
refrefX f
fmm log5.2
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:
X
XVegaX
mf
ffm
,0
,
log5.2
log5.2log5.2
• Colour is defined as the relative flux between two different wavebands, usually written as a difference in magnitudes
Apparent magnitudesApparent magnitudes
Object Apparent
magSun -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
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.
95/465.2/)629( 101010
refrefX f
fmm log5.2
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
name
Central
Wavelength (m)
Bandwidth
(m)
Flux of Vega
(Jy)
U 0.37 0.066 1780
B 0.45 0.094 4000
V 0.55 0.088 3600
R 0.66 0.14 3060
I 0.81 0.15 2420
J 1.25 0.21 1570
H 1.65 0.31 1020
K 2.20 0.39 636
X
XVegaX
mf
ffm
,0
,
log5.2
log5.2log5.2
What is the B-V colour of a source that has a flux proportional to -4?
Band
name
Central
Wavelength (m)
Bandwidth
(m)
Flux of Vega
(Jy)
U 0.37 0.066 1780
B 0.45 0.094 4000
V 0.55 0.088 3600
R 0.66 0.14 3060
I 0.81 0.15 2420
J 1.25 0.21 1570
H 1.65 0.31 1020
K 2.20 0.39 636
X
XVegaX
mf
ffm
,0
,
log5.2
log5.2log5.2
It is also useful to have a measurement of intrinsic brightness that is independent of distance
Absolute Magnitude (M) is therefore defined to be the magnitude a star would have if it were at an arbitrary distance D0=10pc:
The value of m-M is known as the distance modulus.
Absolute magnitudesAbsolute magnitudes
24 r
LF
(note the zeropoints have cancelled)
5pc
log5
pc 10log5
star
star
D
DMm
ExampleExample
Calculate the apparent magnitude of the Sun (absolute magnitude M=4.76) at a distance of 1 Mpc (106 pc)
5pc
log5
starD
Mm
• 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 diagramThe colour-magnitude diagram
Precise parallax measurements allow us to plot a colour-magnitude diagram for nearby stars.
The Hertzsprung-Russel (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:
Absolute magnitude (y-axis) requires measurement of flux and distance
blue
red
blue
red
blue
red
L
L
Lr
Lr
f
f
2
2
4
4
Types of starsTypes of stars
Intrinsically faint stars are more common than luminous stars
Main sequence fittingMain sequence fitting
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.
NGC2437
Main sequence fittingMain sequence fitting
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
Nearby stars (parallax) distant cluster (apparent magnitudes)
Main sequence fittingMain 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:
This gives a distance of 1.4 kpc to NGC2437, reasonably close to the accepted distance of 1.8 kpc.
5log5
7.10
d
MVDM V
BreakBreak
Variable starsVariable 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 starsVariable 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 starsVariable 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 starsVariable 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:
9.4)1(log96.2 PM I
Where the period P is measured in days, and the magnitude is measured in the I band.
Instability stripInstability 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
RR Lyrae
Pulsating whitedwarfs
W Virginis
RR Lyrae StarsRR 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
Abs
olut
e M
agni
tude
Period (days)
Log (Period)
Schematic representation
RR Lyrae variablesRR Lyrae variables
RR Lyrae stars have average absolute magnitudes M=+0.6. How bright are these stars in Andromeda?
Summary: the distance ladderSummary: 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 period-
luminosity 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!
SpectroscopySpectroscopy
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 spectrumExample: the solar spectrum
What elements are present in the Sun?
Solar spectrum
Example: the solar spectrumExample: the solar spectrum
What elements are present in the Sun?
Balmer lines (Hydrogen)
Example: the solar spectrumExample: the solar spectrum
What elements are present in the Sun?
NaD
Example: the solar spectrumExample: the solar spectrum
What elements are present in the Sun?
Ca H+K
Example: the solar spectrumExample: the solar spectrum
So: the Sun is mostly calcium, iron and sodium?? No! Not quite that simple…
Solar spectrum
Stellar spectraStellar spectraStellar spectra show interesting trends as a function of temperature:
Incr
easi
ng t
empe
ratu
re
Spectral classificationSpectral 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 classesCharacteristics of spectral classesSpectral
Type
Colour Temperature
(K)
Main characteristics Example
O Blue-white >25000 Strong HeII absorption (sometimes emission); strong UV continuum
10 Lacertra
B Blue-white 11000-25000 HeI absorption, weak Balmer lines Rigel
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
The HR diagram revisitedThe HR diagram revisited
Henry Norris’ original diagram, showing stellar luminosity as a function of spectral class.
The main sequence is clearly visible
Spectral ClassO B A F G K M
Lu
min
osi
ty
The original HR diagram A modern colour-magnitude diagram