| Describe carefully any two of the following Cepheid-calibrated extragalactic distance indicators. Your account should explain the basis of the technique and how it is used to measure distance. Mention any particular advantages or disadvantages. | |
| (i) The Fundamental Plane; | [5] |
| Basis of the method: an empirical correlation exists relating the radius, surface brightness and velocity dispersion of elliptical galaxies. The surface brightness and velocity dispersion are measured quantities (note: surface brightness measured in magnitudes per square arc second does not depend on distance); you compare the calculated linear size with the measured angular size to determine the distance. (Alternatively, you measure the angular size corresponding to a particular contour of surface brightness, which has the advantage that it reduces the effect of sky brightness on the measurement; this is called the Dn–σ method.) It's clearly an angular size distance, since you compare the measured angular diameter with the calculated linear diameter. | [2] |
| Target objects: large elliptical galaxies (not dwarf ellipticals, which have systematically lower surface brightness). | [0.5] |
| Calibration: by the Virgo Cluster distance as derived from Cepheids. | [0.5] |
| Systematics: it's an empirical correlation, so (since we don't have a theoretical understanding) may well have non-obvious systematics. Dependence on metallicity and/or age is quite likely; may also depend on details of galaxy type (e.g. disky versus boxy). | [1] |
| Advantages: it's a whole-galaxy method, so fairly long-range (the limiting factor is the need for velocity dispersion, which means you have to be able to take a spectrum). Elliptical galaxies are bright and common. Disadvantages: the lack of theoretical understanding is worrying (potential systematics), and the method can't be directly Cepheid calibrated since Cepheids don't occur in ellipticals – you have to assume that the spirals in Virgo aren't systematically offset from the ellipticals (but they might be: the ellipticals are more concentrated in the centre, and the spirals on the outskirts, so if you tend to pick the spirals on the near side of the cluster in preference to those at the back, you will get an offset). | [1] |
| (ii) Type II supernovae; | [5] |
|
Basis of the method: measure the expansion velocity of the supernova ejecta using
spectral line Doppler shift. If you can resolve the expanding SNR, you compare this
with the change in angular size, thereby obtaining the angular size distance. If you can't resolve the SNR, you argue that the luminosity is 4πR2σT4, use the spectrum to deduce the effective temperature, and compare the brightness at two different times to get the ratio of radii. Integrating the expansion velocity over the interval between the two observations gives you the difference in radii; you then compare the ratio and the difference to get R1 and R2 directly. Use these to calculate the luminosity (or apparent magnitude) and compare this with the observed flux (apparent magnitude) to get a luminosity distance. (Note that although you calculate linear radii in this method, you never compare them to angular size – therefore it is not an angular size distance.) |
[2] |
| Target objects: spiral and irregular galaxies, since SNe II arise from massive stars which are not found in elliptical galaxies. (Note that you do need to image the host galaxy – that's where you get the redshift from.) | [0.5] |
| Calibration: not needed (this is a geometric method, therefore a primary indicator). | [0.5] |
| Systematics: messy! The expanding shell is not a very good approximation to a blackbody, and the emission lines at different times may come from different depths within the shell, so you need a detailed model of the shell to interpret the data. If the explosion is significantly non-spherical, so that the radial velocities you measure don't correspond to the lateral velocities that make the shell appear larger, you will introduce another error. | [1] |
| Advantages: SNe II are nearly as bright as SNe Ia, and therefore this is a long-range indicator. It is also in principle a primary indicator (the Hubble Key Project did apply Cepheid calibration, but this was to validate the models of the expanding shell &ndash a check, rather than a necessary step). Disadvantages: mostly the nasty model dependence. Also, these are quite rare events, so it takes time to gather a useful sample. | [1] |
| (iii) Gravitational Lens time delays. | [5] |
| Basis of the method: general-relativistic bending of light from a distant source around an intervening large mass. In this case, the distant source is usually a quasar, and the intervening mass usually a cluster of galaxies. Unless the alignment is exact (in which case you get a ring), the result of this is a double image, and the light path length (= travel time) for the two images will not be identical. Quasars are variable on shortish timescales (months): you shift the light-curves of the two images until they line up, thereby determining the absolute difference in path lengths. From the source-lens-observer geometry, you calculate the ratio of path lengths: this gives you two simultaneous equations, from which you can deduce the total path length for each image. Because you are essentially comparing the angular separation of the source and the lens (which is what determines the relative path lengths) with the absolute length given by the difference in travel times, this is an angular size distance. | [2] |
| Target objects: quasars or other very bright variable sources (you need the variability to determine the time delay: a completely stable source like a normal galaxy won't do the job). | [0.5] |
| Calibration: not needed: geometric, therefore primary. | [0.5] |
| Systematics: nasty. You need a good model of the mass distribution in the lens to calculate the path lengths, and since the lens is generally a cluster of galaxies rather than a single object, this is decidedly non-trivial. Furthermore, if your source is extended (e.g. a double-lobed radio source), the lens may not amplify all parts of it equally, which makes matching up the light curves a bit of a black art. | [1] |
| Advantages: extremely long-range, since the source can be very distant indeed, and independent of Cepheid calibration. Disadvantages: low statistics, long observation times (since the time delay is years, not weeks), and the model dependence discussed above. | [1] |
(2005 Q5(b) [slightly modified].)