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.

M = m -5 log10d + 5

An important equation!

It allows us to connect m and d with M

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.

Measurements of Distance in Astronomy.

Reminder!

Towards Earth

Rotating Spiral galaxy

Tully-Fisher Relationship

Named after R.Brent Tully and J.Richard Fisher

5.9 x 10-6 eV

1420 MHz = 21.1 cm

Nuclear Spin

Electron Spin

Hydrogen ground state is split in energy with the statehaving the proton and electron spins parallel beingslightly higher in energy ( ). Compare thiswith the binding energy of the 1s state = -13.6 eV

5.9 x 10-6 eV

Hydrogen in its ground state will absorb 21 cm. radiation. If it is already in the upperstate it will radiate 21 cm. radiation, which penetrates the dust clouds in interstellarspace. 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 = 42. r3

G.M

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.

M here is mass

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 stellarand 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

Absolute Magnitude

Total Energy emitted

T

Energy emittedper unit area

SIZE Assuming a geometryfor the star.

Standard Candles

There are three requirements:-

a) They must be bright.

b) They must have a well-defined luminosity

c) They must be relatively common.

Remember there is no perfect Standard candle!

M = m -5 log10d + 5

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 = 42. r3

G.M

However the nearest stars are well beyond this distance and if a starhas 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

Mv2 = G.M.m dM r2

Now the period P for an orbit is 2.radius/velocity i.e. v = 2. dM

P

1.

2. Mv2 = M. (2. dM/ P)2

dM

= G.M.m r2

G.(M + m) P2 = 42.r3 = 42 .(M + m)2 .dM

3 = 42 .(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.

dM

Since dM = m .r (M + m)

Stellar Masses

This shows the motion of a binarystar 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 starsseparately 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 = 42. r3

G.(M + m)

Measurement of period and semimajor axis givesus 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 = 42. r3

G.(M + m)

To get separate masses we need moreInformation. Each star moves in an a ellipse about centre-of-mass. Byplotting separate orbits usingbackground 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 = 42. 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 MassesIf we have a binary system of masses M and m at distances D and dthen P2 = 42( M + m )2 D3 for mass M

G.m3

P2 = 42( M + m )2 d3 for mass m

G.M3

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 Shapley- Curtis 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 Localcluster 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 ofgalaxies.The error bars are from measurements of distances made using type 1a supernovae as Standard Candles.

From these resultsH0=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

we also have v = H0. d ----------- Hubble’s law

c

• 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.