Transcript Stellar Masses

Week 6
Brief reminder of last week’s lectures on distance measurements
Measurement of stellar masses
The work of Edwin Hubble
- Establishment of the idea that other galaxies exist.
- Classification of galaxies
- Hubble’s Law
- Allows measurement of distances
- provides basis for idea of an expanding Universe
- hence it provides the basis for the Big Bang Model.
Reminder!
Measurements of Distance in Astronomy.
An important equation!
It allows us to connect m and d with M
M = m -5 log10d + 5
This equation is called Pogson’s equation and its importance lies in
the fact that it allows us to connect M, m and d.
A knowledge of any two of the three allows us to obtain the third of them.
Tully-Fisher Relationship
Towards Earth
 Named after R.Brent Tully and J.Richard Fisher
Nuclear
Spin
Electron
Spin
5.9 x 10-6 eV
1420 MHz
 = 21.1 cm
Hydrogen ground state is split in energy with the state
having the proton and electron spins parallel being
Rotating Spiral galaxy
slightly higher in energy (5.9 x 10-6 eV ). Compare this
with the binding energy of the 1s state = -13.6 eV
Hydrogen in its ground state will absorb 21 cm. radiation. If it is already in the upper
state it will radiate 21 cm. radiation, which penetrates the dust clouds in interstellar
space. 21.1 cm. is in the radio part of the spectrum.
Note:- First observed by H.Ewen and E.M.Purcell (Harvard) 1951
Predicted by H.C.van de Hulst 1944
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
• 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.
• So large mass means large M means small Period P and hence larger
rotational velocity and line broadening.
Tully-Fisher Relation
• The Tully-Fisher relation has been established at many wavelengths
but it is most successful in the infrared since there is much less
absorption and scattering at this
wavelength.
• The picture shows an example
of the relationship between
absolute magnitude and
the measured width not of the
21 cm line but a line in the
infrared part of the spectrum.
We now see that there are a variety of methods of determining stellar
and galactic distances.The table shows the values they give for the
distance to the Virgo cluster and the ranges they cover.
Apparent
Magnitude
Parallax
EM Spectrum
Distance
T
Absolute
Magnitude
Energy emitted
per unit area
Total Energy
emitted
SIZE
Assuming a geometry
for the star.
Standard Candles
There are three requirements:a) They must be bright.
b) They must have a well-defined luminosity
M = m -5 log10d + 5
c) They must be relatively common.
Remember there is no perfect Standard candle!
Problems in measuring Distances
• Zero Point Error:-If we make a mistake in measuring the distance to
a primary indicator then this error is propagated up the chain of
distance measurements[The Cosmological Distance Ladder] since the
value of M for the secondary indicator has been derived assuming the
wrong d.
• Extinction:-of standard candles in galaxies will also cause inaccuracies
It occurs because of matter in interstellar space within the Milky Way
and other galaxies.
Note:-This effect will be much smaller out of the galactic plane.
• No Perfect Standard Candle:-Some of the methods assume peak
magnitudes for various objects-Supernovae, Cepheids etc- but they are
only average magnitudes.Each varies from the mean.As we move to
greater distances it will be easier to detect only those at the high end of
the distribution i.e.the brightest.
This obviously leads to errors and results in what is called the
Malmquist Bias. This leads to their distances being underestimated.
Stellar Masses
• In essence there is only one way of measuring masses and that is
through the gravitational interaction with other bodies.
• If we were able to observe the Solar system from a distance, say several
hundred AU, then we could measure the periods of the planetary orbits
and deduce the solar mass from Kepler’s Third Law.
P2 = 42. r3
G.M
However the nearest stars are well beyond this distance and if a star
has planets we cannot distinguish them.[We are only just beginning to
see a few planetary systems and these are Gas giants like Jupiter]
• Fortunately a large proportion of stars are binary systems and are in
orbit about their common centre-of mass.In general one of the pair does
not dominate the system as in the Solar system.
As a result we have to use the full version of Kepler’s Law.
The Effect of Centre-of-Mass
1.
Mv2 = G.M.m
dM
r2
Now the period P for an orbit is 2.radius/velocity
i.e. v = 2. dM
P
2. Mv2 = M. (2. d / P)2
= G.M.m Since d = m .r
M
M
2
r
dM
(M + m)
dM
P2 = 42.r3
G.(M + m)
= 42 .(M + m)2 .dM3 = 42 .(M + m)2.dm3
G.m3
G.M3
3. In the case of the Earth and Sun the imbalance in mass is such that the
approximation of a static Sun is a good one. However Kepler’s Laws
apply to all systems moving under gravity and often the masses are
closer together so we cannot assume that one of them is fixed.
Stellar Masses
This shows the motion of a binary
star system. The stars move in
elliptical orbits about the common
centre-of-mass.
Note:-Alpha Centauri and Sirius
are examples of binaries. Cases
where we can observe the two stars
separately are best because we can
measure both orbits.
• Distant pairs cannot be resolved but can still be studied. Each star has a
distinctive spectrum. At any given moment they will have different
velocities relative to Earth.This can be determined from the
different Doppler shifts.This allows us to get the full orbital
characteristics if the orientation of the plane of the orbits is known.
P2
=
42. r3
G.(M + m)
Measurement of period and semimajor axis gives
us M + m from Kepler’s Law.
If period is very long this may be difficult.
a can be determined from measurement of
angular separation but we need distance to the
stars which we get by stellar parallax or
spectroscopic parallax.
P2 = 42. r3
G.(M + m)
To get separate masses we need more
Information. Each star moves in an
a ellipse about centre-of-mass. By
plotting separate orbits using
background stars as reference we get
the centre-of-mass
Relative sizes of the two orbits around
C-of-m then gives M/m
Mass – Luminosity Relationship for stars.
Putting many mass and luminosity
measurements together show a clear
relationship between the two.
This now allows us to infer stellar
masses for stars beyond the reach of
parallax measurements
We also conclude that these stars all
belong to a common class of objects
which we will come to see as
Main Sequence Stars.
Added Difficulties
1) Proper motion of binary
2) Ellipse may lie at an angle and we have to allow for that.
Stellar Masses
• If the observer is in the same plane as the stars then the radial velocities
calculated correspond directly to the actual velocity components of the
orbital motions.
If the plane is at an angle to the observer then the measured radial
velocities are lower than the actual velocities.
If it is in the plane of observation then twice per orbit the stars will
pass in front of each other and the total luminosity will dim.[Eclipsing
Binary]
Sufficient visual or eclipsing binaries are known to check the
mass-luminosity relationship derived later.
P2 = 42. r3
G.(M + m)
Gives the sum of the masses provided that the semi-major axis is known.
• This process is made more difficult by the proper motion of the star
as well as the difficulties of stellar orbits being at an angle.
Stellar Masses
If we have a binary system of masses M and m at distances D and d
then
P2 = 42( M + m )2 D3 for mass M
G.m3
P2 = 42( M + m )2 d3
G.M3
for mass m
If we have only the separation of the two stars we get M + m.
If we can measure both orbits we can get both m and M.
The work of Edwin Hubble
• Edwin Hubble( 1889-1953 ) – An American astronomer
made a series of important contributions to Astronomy.
- He resolved the question of whether the spiral nebulae were relatively
small, nearby objects scattered around our galaxy or were separate
“Island Universes” = other galaxies.[This was the so-called ShapleyCurtis debate of the 1920s]
- He classified the observed galaxies[see later]
- He discovered the Law that bears his name. This is the main basis of
the Big Bang model, the current generally accepted model of
Cosmology. It has also become a means of determining distances.
M31-Andromeda Galaxy-2.2Mly from Earth, Part of our Local
cluster of galaxies.
Other galaxies
• Middle of 18th C--Kant and Wright suggested Milky Way was finite in
size and disc shaped. They suggested nebulae might be other “Island
Universes”.
• 1920s-Shapley and Curtis debated whether nebulae were inside or
beyond the Milky Way.
• Hubble( 1923 ) detected Cepheid variables in Andromeda( M31 ).He
was able to determine M from the period-luminosity relation and
measure m. He was then able to determine their distances from
M = m -5 log10d + 5
His original value of 285 kpc is about one-third of present value of
770 kpc.
It was enough to establish that M31 is indeed another galaxy since the
size of the Milky Way was estimated by Shapley to be 20kpc.
Doppler Shift
• If v  c then the shift in wavelength due to the motion of an object
relative to an observer is  = v/c.
• If v is +ve then  increases and we have a redshift, Object looks colder.
If v is +ve then  decreases and we have a blueshift, Object looks hotter.
• If 0 is the wavelength emitted by a stationary source then
 = (  - 0 )
• We now define the Redshift Z as
Z = (  - 0 )/ 0 = / 0 = v/c---provided v << c
• We can measure  spectroscopically. As a result if we can identify a
spectral line and measure  then we can measure its relative velocity.
• If v is not much less than c then we must use the full expression
Z = [(c + v)/(c - v)]1/2 - 1 = / 0
Z = [(c + v)/(c - v)]1/2 - 1 = / 0
Now [1 + x]n = 1 + nx + n(n-1)x2 + n(n-1)(n-2)x3 + ---------2!
3!
Providing x < 1
So ( 1 + Z )2 = (1 + v/c)(1 +v/c + v2/c2 + -----)
= (1 + v/c)2 for v/c << 1
Therefore ( 1 + Z) = ( 1 + v/c)
i.e. Z = v/c = / 0
Hubble’s Law
• The prelude to Hubble’s Law was the establishment of measurements
of galactic distances and also measurements of their Doppler shifts.
• Vesto Slipher [Lowell Observatory] measured the redshifts of many
galaxies[nebulae] in the early decades of the 20thC. They were
expected to be random but he found that they were all moving away
from us and each other although a few such as Andromeda( M31)
were approaching ( blue-shifted by 300 km per sec in the case of
Andromeda). Speculation began about an expanding Universe. It was
in 1925 that Hubble defined the distance to M31 and Slipher
concluded almost all galaxies were moving away from us.
• Hubble continued his search for Cepheid variable stars in other
nebulae and this gave him distances to the galaxies for which Slipher
had measured Doppler shifts.
Hubble’s Law
• Hubble combined his results with Slipher’s and found a clear
correlation between a galaxy’s recessional velocity v and the distance
from the Earth d.
Note: Most of Hubble’s measurements were actually made by his
assistant Milton Humason( 1891-1972 ).
• In 1929 Hubble published a paper entitled
“A relation between Distance and Radial Velocity among
Extra-Galactic Nebulae” at the U.S. National Academy of Sciences.
• The end result is summarised in the simple Law bearing his name
v = H0.d
Usually v is in kms-1,d is in Mpc so H0 is in kms-1Mpc-1
The constant H0 is called Hubble’s constant.
A 1936 version of Hubble’s results published by him.
Hubble’s Law
Figure shows the distances and recessional velocities for a sample of
galaxies.The error bars are from measurements of distances made using
type 1a supernovae
as Standard
Candles.
From these results
H0=65 kms-1Mpc-1
Distances with the Hubble Law.
• Hubble’s law was derived from measurements with the various
methods for measuring distances we examined earlier.
• Now we can turn it around and use it as a measure of distance.
• We can measure the Redshift z and if v  c we can write
z=v
c
we also have
v = H0 . d
----------- Hubble’s law
• Putting them together
d = z.c
H0
And we have a method of measuring distance over a very large
range of distances if we assume that it holds at all distances.