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