2 Radiation from the galaxies
Stars occur in great collections called galaxies. The distribution and motion of galaxies provide us with the first important experimental information on which we shall build our understanding of the type of universe we inhabit. So, what do we know about galaxies?
All the stars that can be distinguished by the naked eye – a few thousand in number – belong to one galaxy: our own Milky Way Galaxy. Sometimes it is just written Galaxy, with a capital G, to distinguish it from all the billions of other galaxies in the observable Universe.
Our Galaxy has the overall shape of a flattened spiral. As shown in Figure 1, we are halfway inside the Galaxy so the spiral form is not obvious to us as we look at the sky. In fact it has only been revealed by detailed radio mapping. This is because there are large sections of the Galaxy that are obscured from optical observations by the intervening dust and gas which scatter and absorb visible light. (However, they are essentially transparent to radio waves.) If there were an unobscured view through telescopes, we would be able to see to the far side of the Galaxy, and in doing so we would record about 1011 stars within it.
Fortunately for astronomers, the stars, though vast in number, display a degree of uniformity which makes their classification and study possible. Their masses are broadly similar – most are contained within a couple of orders of magnitude, with our Sun having a typical average mass. They do differ considerably in size and therefore in density (since their masses are similar), but these differences are thought principally to relate to different stages in the life of a star rather than to different types of star. These various stages correspond to different types of nuclei taking part in the nuclear reactions that provide the energy output. It is one of the features of astrophysics that one is able to take nuclear data obtained in the laboratory and use them to understand the various stages of evolution seen in different stars. The main process in young stars is a sequence of reactions leading to the fusion of hydrogen nuclei to make helium nuclei. Later, helium fuses to form carbon, then carbon undergoes reactions which build up heavier nuclei. This can continue up to iron. The different reactions take place at different temperatures and pressures, so this progression of reactions governs the sequence in which a star changes its size and appearance.
Stellar evolution is too slow a process for us to see any particular star undergo change in one human lifetime, apart from a few exceptionally young stars, and some very old stars undergoing gravitational collapse leading to supernova explosions. But by observing different stars at their respective different stages of development, it is possible to piece together the whole of a typical stellar life cycle.
The general idea of an evolutionary sequence – one that can be reconstructed on a computer – is of concern in this course for the following reason: when astronomers look at a very distant galaxy, they are receiving radiation that left its source long ago. The galaxy will, therefore, seem younger than it actually is now. The travel time for the light may have been hundreds or thousands of millions of years. To interpret the observations, one needs to know how the power output of the galaxy evolves with time.
The light from a typical galaxy derives mostly from the stars it contains, with only a small amount from interstellar matter. So, to understand the power output of a galaxy one has to add together the light from about 1011 stars at various stages of development. We cannot assume that the stars of the distant galaxy will be emitting the same amount of light as those of our own Galaxy today; the distant stars will be seen at an earlier stage in their development when perhaps their power output was different from what it actually is now. Indeed, there is an added complication. The evolution of a star depends critically on its mass. A very massive star will shine much more brightly than a less massive one, but over a much shorter time period. It could be that when we look at a young galaxy, we see many more massive stars living out their brief active lives than we observe today in our own Galaxy. So the mass distribution of active stars in our Galaxy might not be representative of what we see happening in the distant younger galaxy. It is difficult to know how to compensate for this. The mass distribution of the stars in a galaxy depends on the way the galaxy was formed, and unfortunately, the formation and growth of galaxies remains an unsolved, or at least poorly understood, problem.
In summary, we know little about the way the power output of a galaxy changes with time, and this represents a severe limitation on the cosmologist's use and interpretation of astronomical data on power output. The frustration this causes will become apparent later.
The light from a galaxy can tell us more than just its overall power output. Additional information comes from the analysis of its spectrum. Let us assume that light comes mainly from the stars and that we can ignore interstellar matter. The majority of galaxies are too far away for it to be possible to resolve individual stars, and therefore the best one can do is to take the spectrum of the galaxy as a whole: 1011 stars summed together. How can we relate this to a stellar spectrum?
The light from a star comes from the hot layers of gas near its surface. This light filters out through the dilute layers above. As it does so, characteristic patterns of absorption lines are imprinted on the spectrum.
Figure 2 shows part of the absorption spectrum of the Sun, compared to an emission spectrum produced in the laboratory.
A beam of white light is passed through a bulb containing calcium vapour. Explain, in broad terms, with the aid of diagrams, the nature of the light transmitted, (a) in the direction of the beam, and (b) at an angle to it.
(a) At position A (Figure 3), the spectrum of the observed light consists of a flat continuum with absorption lines (corresponding to the transitions between the ground state and other states) as in Figure 4(a).
(b) At position B, no direct light is seen. The light that is observed comes from calcium atoms that have been excited, and is therefore an emission spectrum of lines on a dark background, with the wavelengths of the lines the same as those of the absorption lines seen at position A. This is shown in Figure 4(b).
The spectral lines in the light from a galaxy enable the astronomer to identify the elements emitting the light. Any of the ninety or so stable elements may be present, but the lighter elements, especially hydrogen, are usually the more abundant. Because some elements are common to most stars, the absorption lines of these particular elements will be visible in the light of the galaxy as a whole. Also, the absorption lines of magnesium, sodium, calcium and iron are often easy to distinguish, even though other elements, such as hydrogen and helium, are more abundant than these.
Usually the absorption lines are sharp and identifiable, despite several effects which can broaden them. One of these effects is a Doppler shift caused by the rotation of the source. For example, the spectral lines of calcium atoms moving towards us would have their apparent frequencies systematically increased (blueshifted) according to the equation
where f0 is the observed frequency, f1 is the emitted frequency and is the speed of appoach. The spectral lines of atoms moving away from us would be correspondingly redshifted. This Doppler shift causes the width of a given spectral line to broaden if we are viewing the rotating galaxy along the plane of its disc so that the light-absorbing atoms belonging to stars on one side are moving away from us, while those on the opposite side are moving towards us (refer back to Figure 1). In addition, the star itself might be rotating so that different parts of the star would have different components of velocity towards or away from us, thereby increasing the broadening. Random motions of stars can also cause line broadening in a similar way.
The shifts and broadenings of the lines are not usually bad enough to mask the spectrum. Therefore, although a galaxy is a very complicated light source, its light is not just a meaningless jumble of fuzzy lines. There are some features, such as the calcium lines, which stand out sharply.
(a) It can be shown that when radiation rises a height H near the Earth's surface, there is a shift in frequency towards the red end of the spectrum given by Δf/f = −gH/c2, where g is the acceleration due to gravity. When radiation starting from the surface of a body of mass M and radius R escapes to a large distance, it suffers a fractional frequency shift Δf/f = −GM/Rc2, where G is the gravitational constant. Outline the steps you would need to take to establish the connection between these two statements without necessarily giving the derivation. (Hint: Bear in mind that g = GM/R2.)
(b) Imagine a star with a mass M = 2 × 1030 kg, a radius R = 7 × 108 m and a period of rotation T = 2 × 106 s (similar to the Sun). Suppose also that the typical speed of turbulent motion in the atmosphere of the star is 6000 m s−1. How is the frequency of the hydrogen line whose wavelength in the laboratory is 656 nm, affected by:
ii.Doppler shift due to rotation;
iii.Doppler shift due to turbulence?
(The values of G and c are given in the data sheet, and the formula for the Doppler shift of light is given in the equation above.)
(a) The result Δf/f = −gH/c2 applies only to small changes in height, H. For a large change, one must take account of the fact that g changes with height. The value of g is given by equating the weight of a body, mg, to the gravitational attraction. Thus at the surface of the Earth
where M and R are the mass and radius of the Earth respectively. At any distance r (greater than R), we may write more generally
At each distance r, the frequency shift caused by rising to r + dr is given by the formula .
where we write dr in place of H. Thus the total frequency shift suffered in escaping to a very large distance (‘infinity’) is determined by:
where f1 is the emitted frequency and f0 is the observed frequency. This gives the method asked for in the Question.
Performing the integration (which the Question did not ask for) gives the stated result. In fact,
Provided the magnitude of Δf = f0 − f1 ≪ f1, we can approximate the left-hand side of this equation by
and conclude that, for any emitted frequency f.
(b) (i) Using the result of part (a), we can find the effect of the gravitational redshift on light leaving the star as follows:
All the radiation leaving the star has been ‘redshifted’ by this amount by the time it reaches the Earth, which we may certainly regard as at an ‘infinite’ distance from the star.
(ii) The speed at the edge of the star is
The Doppler shift is
when is a speed of approach. Here,
This could be used directly in the formula but since /c is so small, we can use the approximation
so the fractional change in frequency, Δf/f, is simply:
Light from the advancing limb is blueshifted by this amount, and from the receding limb is redshifted by this amount. The width imparted to lines is thus the fraction 14.7 ×10−6 of their central frequency.
(iii) The turbulent motion again produces both blueshifts and redshifts, of amount ±6000 m s−1/c = ±20 × 10−6.
The frequency of the 656 nm line is given by
The frequency shifts are therefore as shown in Table 1.
|gravitation||−2.12 × 10−6||−970|
|rotation||±7.33 × 10−6||±3350|
|turbulence||±20 × 10−6||±9150|
In many galaxies, spectral lines have been observed whose relative positions correspond exactly with those of a known element (so that the spectrum is confidently identified), but whose absolute frequencies are all noticeably shifted. This can have nothing to do with the random motions of the stars within the galaxy, or the rotations of either the individual stars or of the galaxy as a whole; all these motions (being as often towards us as away from us) would merely broaden the line. A net shift of all the lines in a spectrum seems to suggest that the galaxy itself has a line-of-sight motion (i.e. a component of velocity towards or away from us). The shift is nearly always to longer wavelengths, i.e. towards the red end of the visible spectrum, so it is referred to as the ‘redshift’. If the redshift is interpreted as a Doppler shift (an interpretation we shall reconsider later) due to the motion of the galaxies, then it follows that most of the galaxies are receding from us. In other words, the Universe is expanding.
The term ‘redshift’ has a precise definition: it is equal to Δλ/λ, where Δλ is the shift in wavelength of a line whose emitted wavelength is λ. The value of Δλ/λ is the same for all the lines in the spectrum of an object receding at a given speed, and is normally denoted by z:
where λ0 is the observed wavelength now, and λ1 is the wavelength at emission. These redshifts in the spectra of galaxies are generally far larger than the stellar effects considered in Question 2, as can be judged from Figure 5.
In summary then, when dealing with light from stars, there are three ways in which the frequency can be shifted:
(i) The Doppler shift due to motion – whether that arises through the rotation of the star, its bodily motion along the line of sight, or turbulence in its atmosphere. This type of shift, which can be red or blue, is accounted for by special relativity.
(ii) The gravitational redshift, arising from general relativity.
(iii) A new type of redshift due to the recession of the galaxies, which is also a consequence of general relativity.
(Note that it is customary to reserve the suffix, 0, for the values of quantities as they are at the present time, t0. Values at other times must carry some other suffix. In this way, we end up with the rather counterintuitive situation where λ1 applies to an earlier time, t1, than λ0.)
Hydrogen has emission lines at 434, 486 and 656 nm. A galaxy is observed to have these three lines redshifted in wavelength by 2% (i.e. z = Δλ/λ = 0.02). What wavelengths will be observed? What is the frequency shift Δf/f ?
If λ1 is the emitted wavelength and λ0 is the observed wavelength, then
Thus the observed wavelengths are:
The relative frequency shift is given by
Note that the shift in frequency is the same for all lines of the spectrum, but differs from the shift in wavelength, which is z.
As we shall see later in this course, the redshift in galactic light has provided one of the main clues to the nature of the large-scale structure of spacetime. But to discover how the pattern of the redshifts observed in different galaxies reveals this structure, we need further information about the distances of galaxies.