Transcript No Slide Title

Last Week
1. Celestial Co-ordinates
2. Measurements of Distance to add to Surface Temperature and Mass
3. Locally we can use Stellar parallax.
4. To go to greater distances we need to use measurements of brightness.
Stellar Parallax
1AU
Star
Tan p = p =1AU / d
 1 radian = 2.063 x 106 seconds
 If we define 1 parsec = 2.063 x 106 seconds – Unit of distance
 d = 1/p where d is in parsecs and p is in arcseconds.
Parallax
From Earth we can only use this technique for about 10,000 stars.
HIPPARCOS (High Precision Parallax Collecting Satellite)
- was launched by ESA in 1989.
- measured parallax for 118,000 stars to 1-2 milliarcseconds to
an error of 10%.
- This is for all stars within several hundred parsecs. In other words
the method has been applied for stars out to about 100pc or 300 ly.
This sounds a large number but remember the numbers of stars in
typical galaxies - 1011
Units of distance
Astronomical Unit = Mean Earth-Sun distance = 149.6 million km.
Light year = distance travelled in vacuum by light in 1 year.
= 9.46 x 1012 km
= 63,240 AU
Parsec = is the distance from which the Earth’s orbit subtends
an angle of 1//. (1 arcsecond)
1 pc = 3.26 ly = 3.086 x 1016 m = 206,280 AU.
Note:- kiloparsecs, Megaparsecs and gigaparsecs are all in use.
Distance and Brightness
Stellar parallax is a reasonably secure method of measuring distance
but it is limited to a relatively small number of stars close to us.
We need other methods of measuring distances to stars. Many of them
rely on measurements of brightness. We must now consider
brightness before we can look at other measures of distance
Luminosity = power radiated by a star.
Stellar Brightness
• Among the most basic observations are distance, brightness(luminosity),
surface temperature, mass etc.
• Luminosity- once we know the distance of a star we can determine the
luminosity, a measure of the total power emitted by the star.
2R
Star radiates isotropically (reasonable assumption). The
energy is spread evenly over the surface of a concentric
spherical shell centred on the star. Now the surface of a
sphere is 4R2 so the flux at any point on the concentric
shell is inversely  R2, the distance from the star.
• If a star’s apparent intensity, how bright it appears from the Earth, can
be measured and its distance is known then its absolute intensity can be
calculated.
Stellar Brightness
• Stellar separations and intensities vary over many orders-of magnitude.
As a result it is convenient to use logarithmic scales.
• Astronomers use relative measures of Intensity.
The system is based on the assumption that iVEGA = 1.0 and the
apparent intensities of all other stars (i) are measured relative to the
intensity of Vega. We define the apparent magnitude (m) of a star as
m = -2.5log10 i ----- definition.
Here m is related to how bright the star appears in the night sky.
Brightness
 Historically Astronomers used the Magnitude Scale for Brightness.
1ST magnitude appeared twice as bright as 2nd magnitude
- Hipparchus could distinguish down to 6th magnitude.
This is the quantity m = apparent magnitude
- This is how bright an object appears from Earth.
- Note that m gets larger for objects that are fainter.
Note that 100 = (2.512)5
We define m = -2.5log10 i and iVEGA = 1.0
Real energy output of a star = Luminosity = Absolute Magnitude (M)
We relate m and M. They are equal at a distance of 10pc.
Stellar Brightness
m = -2.5log10 i ----- definition Apparent Magnitude
• Since iVEGA = 1.0 it has m = 0.0
There are a few stars brighter than Vega in the sky and they all have -ve
apparent magnitudes. The brightest of all is Sirius at m = -1.5.
An object ten times brighter than Vega has m = -2.5
• One peculiarity of this system is that dimmer stars have larger apparent
magnitudes. Thus i = 0.1 has m = + 2.5
10 times dimmer than Vega
i = 0.01 has m = +5.0 100 times dimmer than Vega
Note:-Eye can detect stars to m = 6 or 7. [System due to Hipparchus.]
Stellar Brightness
• Conversion to Absolute Magnitude( M ):- If we have an absolute
intensity (I ) then i  I
i = C.I
d2
where C is a constant of proportionality.
So
i d2 = C.I
taking logarithms to base 10 and multiplying by -2.5 gives
-2.5 log10 id2 = -2.5 log10CI
-2.5 log10i -2.5log10d2 = -2.5log10C -2.5log10I
If we then write M = -2.5log10I as for apparent magnitude then we
can write
m -5log10d = M - 2.5log10C
To connect m and M we must decide on the constant term.
Stellar Brightness
We do this with the following arbitarary definition:M = m when the star is viewed from a distance d = 10 pc.
Then M = m -5 log10d + 5
We now have a link between M,m and d where d is in parsecs.
[Note: we have assumed that the inverse square law is the only
reason for the dimming of the light from the star.It takes no account
of any possible absorption in dust between us and the other star.]
[Note: This not an SI system of units]
How can we use this equation?
Example:- From the Earth the Sun is 4.8 x 1010 times brighter than Vega.
What is the Sun’s absolute magnitude given that Vega is 8.1 pc from
Earth?
Example:- From the Earth the Sun is 4.8 x 1010 times brighter than Vega.
What is the Sun’s absolute magnitude given that Vega is 8.1 pc from
Earth?
i = 1.0 for Vega and 4.8 x 1010 for the Sun - Apparent magnitudes
mSUN = -2.5 log (4.8 x 1010 ) = -26.7
Now 1pc = 2.06 x 105 AU so for Sun d = (1/ 2.06 x 105) pc
Therefore MSUN = mSUN -5log10(1/ 2.06 x 105 ) + 5 = +4.9
What is the Absolute magnitude for Vega?
MVEGA = mVEGA - 5log10dVEGA + 5
= 0 - 5log10( 8.1 ) + 5
= 0.5
Hence since M = -2.5 log10I and 0.5 = -2.5 log10IVEGA , 4.9 = -2.5 log10IS
4.9 - 0.5 = -2.5( log10ISUN - log10IVEGA )
So IVEGA = 10(4.9 - 0.5)/2.5.ISUN = 57 ISUN
Pogson’s Equation
 iVEGA = 1.0
------ definition
 m = -2.5log10 i ----- definition.
These definitions are just
something you have to accept
 M = -2.5log10I -----definition
M = m -5 log10d + 5
----Pogson’s Equation
M = m when the star is viewed from a distance d = 10 pc
This gives us a link between distance d in parsecs and the apparent
(m) and absolute (M) magnitudes.
Stellar Sizes
Absolute Magnitude(M) is a measure of the total power that is radiated by
an object.
From the star’s spectral type we obtain the surface Temperature T.This
gives via the Stefan-Boltzmann Law
P = .T4
the amount of radiation emitted per unit surface area.
It is then a simple matter to obtain a measure of the size of a star from
Surface area = Total Power/ P
If we assume that the star is spherical then surface area is 4R2 and we
can deduce the radius of the star.
On this assumption R2 = Total Power/ 4P
Apparent
Magnitude
Parallax
EM Spectrum
Distance
T
Absolute
Magnitude
Energy emitted
per unit area
Total Energy
emitted
SIZE
Assuming a geometry
for the star.
Stellar Distances
• We have already seen that the method of Parallax only works for
nearby stars. We need other methods.
• To go to much greater distances we need some other kind of yardstick
.
We introduce the idea of the Standard Candle = an object whose
absolute intensity we assume we know. From the observed intensity,
the apparent magnitude m, we can then deduce the distance d from
M = m -5 log10d + 5
Cepheid Variables
• Amongst the variable stars we find Cepheids, named after  Cephei.
The latter was discovered by a young man, John Goodricke, living in
York, in 1784.
Their use to measure distance derives from the work of Henrietta
Leavitt (Harvard), who observed a correlation between brightness and
the period of pulsation (P) for these stars.
It is thought that all Main Sequence stars pass through this phase at
some point in their life cycle.
For Classical Cepheids M = -2.8 logP, where M = average absolute M.
The range of absolute magnitudes for such stars is M = -1 to -7 and
their periods vary from 2-100 days.
The use of these stars as Standard Candles allows distances up to Mpc
to be measured.
Cepheid Variables
This shows the details of how  Cephei
varies with time.
At the top the star’s magnitude varies with
time in days.--The Light curve
Then follows on the same time scale
- the radial velocity
- the star’s surface temperature
- the star’s diameter.
M = -2.8 logP allows measurements to
about 20 Mpc
Cepheid variables
This shows the relationship
between absolute magnitude
and period of intensity variation
for pop.1 And pop. 2 Cepheid
variable stars.
M
Metal-rich
Population 1 And population 2
stars can be distinguished by
their spectra.Type 1 are metalrich and type 11 are metal-poor.
Metal-poor
P
In terms of measuring distance
we can measure their period and
then use this graph to determine
M.A measure of m then gives d
from M = m -5 log10d + 5
Example from “UNIVERSE”-Freedman and Kaufmann
Galaxy IC 4182 Hubble Space Telescope used to make 20 separate
images of galaxy. They found 27 Cepheids in IC 4182.
They were able to plot their light curves. One particular cepheid had
P = 42 days and m = 22.0.
From the spectrum they could tell it was metal-rich, population 1.
They now used the graph of M versus P to find M = -6.5
M = m -5 log10d + 5
m-M = 28.5
We can rewrite our eqn as d = 10(m-M+5)/5 parsecs
So for this case d = 5Mpc
In 1937 there was a Type1a supernova in IC 4182 with m = +8.6
Since m-M = 28.5 for this galaxy we know
M = -19.9 for the supernova.
Type1a are exploding white dwarfs. They are thought to always have
same M. Measurements like this calibrate them as Standard Candles.
Distance Measurements
• It was Hubble’s observation of Cepheid variables in Andromeda which
allowed him to determine how far away it is and establish it as
another galaxy.The use of Cepheid variables is the first step in a chain
of measurements of galactic distances. Beyond tens of Mpc we need
brighter objects -brighter Standard Candles.
• NOVAE
These are thought to be close binary systems in which one component
is a White Dwarf star.The other
component is a giant star which
Artist’s idea of
has grown so large that the outer
R Acquarii-A
layers of H are dragged off by the
nova seen in
grav. field of the White Dwarf.The H
inset with the
builds up on the surface and when
uncorrected
there is sufficient material,sufficient
HST
pressure and temperature we get
a thermonuclear explosion.This is
a NOVA--a new star.
Novae
White Dwarf star is the remnant of stars up to about 8 solar masses
In essence it is the core of a star like the Sun that has shed its outer
layers in a planetary nebula. A typical example of which we see below.
Initially it is quite hot since it was the core of a star but slowly cools
with time.
Typically it has a radius 1% of the solar radius and T = 24,000 K
Helix Nebula
An example of a
planetary nebula
Sirius B is an example
of a white dwarf
Novae
The White dwarf is small in radius but high in density.
It is maintained by electron degeneracy pressure
No White Dwarf can have a mass greater than 1.4 solar masses
- the Chandrasekhar limit. Theory is backed by experiment.
If isolated a white dwarf will simply cool with time since it has no
source of fuel. It will become a cold, black dwarf.
Its size will remain the same since the electron degeneracy pressure
keeps it stable against gravitational pressure.
Novae
The situation is different if it is in a close binary system.
If its companion is a main sequence or Red giant star then it can gain
mass from its companion.
Clump of mass has some small angular velocity. Law of conservation
of angular momentummeans it moves faster and faster as it falls in.
The infalling matter forms an accretion disc – a whirlpool like disc of
matter around the White Dwarf.
This is a new source of energy for the star. The disc becomes hot
because grav.P.E. has been turned into K.E.
It can shine brightly with UV and X-rays.
Novae
More dramatic events can follow.
H from the companion gradually spirals in through the disc and falls on
the surface. The strong gravitational field compresses this gas into a thin
surface layer. Both pressure and temperature rise as the layer builds up.
When T approaches 107 K hydrogen fusion erupts.
The thermonuclear explosion causes the system to shine for a few
glorious weeks.
It generates heat, light and ejects most of the accreted mass.
Accretion now resumes and the process is repeated.
The time between novae depends on the mass of the White Dwarf.
Period ranges from decades to 10,000 years.
Novae
Theory cannot tell whether mass of the white dwarf increases or
decreases with time. In some cases the mass approaches the limit
of 1.4 solar masses.
Its interior temperature rises high enough for carbon fusion to begin.
It ignites almost simultaneously throughout the star and it explodes
completely in what is called a white dwarf supernova or
Type 1 supernova.
The peak luminosity is about 1010 solar luminosity, which fades
steadily.
The spectrum has no H lines ((or almost none) since the White dwarf
has very little H.
NOVAE
The two stars orbit the c-of-m.
As the Red Giant grows some of the
H envelope goes beyond the c-of-m
and it is attracted to the White
Dwarf and accumulates there.
With enough H the T and P is
enough to restart nuclear reactions
on the surface of the star.The rapid
rise in the T causes a huge increase
in intensity and material is thrown
into space.
This does not destroy the two stars
and the process starts again.
NOVAE
• Statistically we expect about 10-40 novae per year in the Milky Way
but we only see about 2-3 because of the vast amounts of interstellar
dust and gas within our galaxy.
• Novae as Standard Candles--the absolute power output varies quite a
lot but it turns out we can still use novae for this purpose.
Novae as Standard candles
a) They are very bright-absolute power output rises 8-10 orders-ofmagnitude.
b) It then decays with a time ( = time to decline by 3 orders-ofmagnitude.) There is a distinct relationship, determined empirically,
between MV ,the peak absolute magnitude, and .
MV = -2 + 2log 
If we measure m and obtain M from the above then we can
use
M = m -5 log10d + 5
to obtain d
NOVAE
Why does the relationship hold?
If you have a smaller, more massive white dwarf
--greater compression of gas on the surface
--greater heating of accreted gases on surface
So a thermonuclear runaway is initiated with smaller accumulated mass
The less massive surface layers are more readily ejected and so the
nova declines more rapidly.
Hence the relationship
MV = -2 + 2log 
The brightest novae are about as luminous as the brightest Cepheids,so
their use spans the same range of distance out to about 20 Mpc.
Stellar Distances
• We can see Parallax, Cepheid variables and Novae as PRIMARY
DISTANCE INDICATORS.
• To measure greater distances we must use SECONDARY
INDICATORS
All such indicators must be calibrated by their observation in galaxies
at known distances. In essence for each such method we try to make
use of the brightest objects in a galaxy.
Next Secondary Indicator
- Ionisation Nebula
or
HII region
Remember we are always hunting for some class of objects
where we believe we know the absolute magnitude M
Levels and Transitions in Hydrogen
The figure shows the levels
in the hydrogen atom.
The zero of energy is at infinity.
The levels are labelled by the
Principal Quantum Number n
and by the energy[on the left]
The series of spectral lines that
were found empirically by various
experimenters are shown.
Each series ends on a particular
level.
656 nm
Stellar spectra-4
Here we see the atomic spectra
for white light,sunlight and a
series of elements.
Note that the last spectra are for
Na in emission and absorption.
These spectra provide clear
fingerprints for the chemical
elements.
656 nm line in hydrogen
Stellar Distances – Secondary Indicators
Orion Nebula –bright young stars at its heart. Lanes of dark dust. The
pervasive glowing red hydrogen gas. Blue tinted dust reflects
light from newborn stars.
Stellar Distances – Secondary Indicators
Hot massive stars live fast and die
young. As a result they do not
move far from the clouds of gas
and dust from which they form.
We find them in star clusters in
the clouds of molecular dust.
Orion is a good example.
The wispy clouds are ionisation Nebulae = HII regions
The gas is irradiated by neighbouring hot stars. The H atoms are excited
and ionised by the radiation. They emit light. In particular they emit
The 656 nm line in the Balmer series [n = 3 n =2]
The H appears red because of the copious emission of this line.
Note the [n = 2 n = 1] line is also emitted but it is in the ultraviolet.
Stellar Distances – Secondary Indicators
One possible standard candle is the average of the three strongest H11
regions in a galaxy. Such regions contain up to 109 solar masses of
ionised H. Measurements of the angular sizes of the H11 regions and
the apparent magnitude of the galaxy allows comparison with galaxies
at known distances. However the regions are irregular. [use with caution]
Stellar Distances
• Brightest Red Supergiants:-seem to have the same absolute magnitude
in all galaxies,as concluded from careful studies by Roberta Humphreys
It seems that mass loss pares the brightest Red Giants down to about the
same maximum mass.Thus they have about the same luminosity.
Sampling a number of galaxies,the average visual magnitude of the
three brightest red stars was found to be
MV = -8.0
• Because individual stars have to be resolved to use this method it has
a slightly larger range than cepheids. Because they are brighter.
we can see them out to  50Mly.
• One problem with this method is that the limited sampling inherent
in such methods can lead to errors.Clearly it is statistically more secure
to take as large a number of objects as possible into our sample.
Globular Cluster M10 – 16,000 ly away in ORPHIUCUM
Diameter is  70 ly Mainly post-Main sequence stars
Globular Clusters
These spherical “clouds” of stars or clumps of stars contain typically
thousands of stars in a relatively small volume. When we look at the
Globular clusters in the Milky Way we find that they have a very large
range in luminosity. They are bright or not depending on how many
stars they have. We can draw the kind
of graph we see here. Other galaxies
give similar curves.
Curve for Virgo
.
Stellar Distances
• Global Cluster Luminosity Function:Statistically it is better to
use a whole class of objects
associated with a galaxy.
One possibility is the use of
Globular Clusters.The picture
shows the variation in magn.
for the globular clusters
around four giant elliptical
galaxies in the Virgo Cluster.
The distribution is well-fitted
by a Gaussian function with
a turn-over magnitude at -6.5.
This can be used as a
Standard Candle
[Can be used out to 150Mly]
Supernovae as Standard Candles
• The brightest supernovae reach M = -19 at the peak of their output. In
theory they can be seen up to 8000 Mly from Earth. This makes them
potentially very interesting as Standard Candles.
• Supernovae are divided into various types.
Type 1 are thought to be formed by the same kind of event that causes
novae. In this case however the White Dwarf collects so much material
that it collapses under the mass of material. The result is a large
explosion, ripping the star to pieces.
• Type 1 can be subdivided and it is only type 1a which is useful because
the others show irregularities in their light curves and spectra.they are
also fainter and rarer.
• Type 1a have a well defined peak magnitude because the mass limit for
gravitational collapse is sharply defined [Chandrasekhar limit of about
one solar mass].
Supernovae as Standard Candles and Rulers
• Typically Type 1a reaches M = -19 with the rise occurring at 0.25-0.5
magnitudes per day. After approx. a week at maximum brightness the
decline begins. Initially it declines at 1 magnitude per week but slows
down to 1/10 weeks.
•
As well as providing a Standard Candle which
r
can be used out to great distances it can also be
used as a standard ruler.
2
The explosion produces an expanding envelope of
d
luminous gas which can be studied
spectroscopically to give Doppler shifts.The
angular expansion of the remnant can also be
used to give the distance from
r = d.2
From the arc we can get its size.
Tully-Fisher Relationship
• 1977-R.Brent Tully and J.Richard Fisher discovered a relationship
between the Doppler broadening of the H 21 cm line and the
luminosity of a spiral galaxy.This provides the basis of a method
of measuring distances to spiral galaxies.
If a spiral galaxy is rotating then it may have a
velocity relative to us overall but the rotational
motion of the galaxy means that the velocity at the
two extremes is different.As a result when we look
at the 21 cm line from this galaxy it will be
broadened since the wavelength will be shifted by
different amounts from the two extremes.
• Tully and Fisher measured the width of the 21 cm line of neutral H
in the radio spectrum for a set of spiral galaxies.Typically the line
shows a double peak.
Tully-Fisher Relationship
• Tully and Fisher showed that there is a correlation between a spiral
galaxies luminosity and its maximum rotational velocity( and hence
the spread in the line width).
• Why does the relationship occur?
The speed of rotation is,of course, related to its mass by Kepler’s
Third Law
P2 = 42. r3
G.M
M here is mass
i.e. larger M means smaller P.
• The more massive the galaxy the more stars it contains and hence
the brighter it is. That is large mass means larger Absolute magnitude.
• Line widths can be measured very precisely so this relationship can
be used to determine the luminosity of distant spiral galaxies.Combining
this with measures of apparent brightness gives us the distance from
M = m -5 log10d + 5
We now see that there are a variety of methods of determining stellar
and galactic distances.The table shows the ranges they cover.
 Parallax d = 1/p . Smallest p = 0.001arcseconds equivalent to  3kly
Cepheid variables M  -6 so Max. d  20Mly or 30 Mly with HST
Novae – depends on mass but we can use them out to 20-30Mly
Red Supergiants M  -8 so Max. d  50Mly
Blue Supergiants M  -9 so Max. d  80Mly
Beyond 80Mly we can no longer distinguish individual stars.
Brightest globular clusters M  -10 so Max. d  130Mly
Brightest HII regions M  -12 so Max. d  300Mly
Tully-Fisher relationship ~ 300 Mly or so.
Type 1A Supernovae M  -19 so Max. d  8000Mly