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Survey, which measured the positions and redshifts (a cosmological unit of distance) of  millions of galaxies.  The Virgo cluster is so close to us that it doesn't all fit even on  a Schmidt plate \ -- \ this picture only shows the centre. So in this  exercise we will concentrate on a different cluster, Hydra I  (also known as Abell 1060, being the 1060th cluster in Abell's catalogue  of clusters). We have a plate of this, called Abell 1060, or J7442 or  10:34-27:22. Get it out on the light table and have a look. The cluster  of galaxies is obvious in the centre.  Hydra I is a cluster very similar to Virgo. It is about the same size,  same mass, and has a similar number of galaxies. However, it is much further  away, as is obvious from the plate. This means the whole cluster fits  comfortably on one Schmidt plate.  \subsection{Distance to Hydra I}  In the first half of this exercise, we will estimate the distance to  the Hydra I cluster. Getting distances is always the most difficult  exercise in astronomy; in many ways it resembles a black art more than  a science. We can't take a ruler to anything outside our Solar System  (our rockets are much too slow), and triangulating only works to  distances of about 20 parsecs, so we have to use guesswork. The method  is this; we assume that we know some property of a distant object as  it really is. For example, we might assume that we know how bright a  particular distant star is. Then we measure how bright it appears from  the Earth. The ratio of the two gives us the distance to the star.  The method we will use in this exercise is galaxy diameters. If we  knew how big a galaxy really is, and could measure how big it appears,  by simple trigonometry we could work out how far away it is. We can  measure the apparent size of the galaxies in Hydra I using the  magnifying eyepiece with a measuring scale. Try measuring the size of  a few. The plate scale of all the plates is $1.12$ arcmin per mm. So  if you measure a galaxy as being 2mm long, its apparent size will be  the angle $2.24$ arcmin.  So we can measure the apparent size of galaxies in the cluster. But  what is their real size? How big is a galaxy? We only really know the  size of one galaxy at all well, our own, the Milky Way. Here's where  we make our first whopping great assumption. Why don't we assume that  the one galaxy we know anything about, the Milky Way, is perfectly  typical? So let us assume that all the galaxies in Hydra I are just  like ours; they have radii of 10 kpc.  So pick a bunch of galaxies in Hydra I. Measure their  diameters. Some of them will be nearly edge-on, but that shouldn't stop  you measuring them. Multiply by the plate scale to get their angular  size. Now work out how far away they are! You can get an answer for  every galaxy you measure, and they won't all be the same. Throw out any  ludicrously discrepant values, and average the rest to get your best guess  of the distance to Hydra I.  One big problem is measuring the galaxy sizes. After all, galaxies don't  have sharp edges, they just fade gently out. Where then do you measure  the edge? There isn't really a good answer. The most you can do is be  self-consistent.  What distance do you get? Current best estimates lie in the range 35 --- 80  Mpc, but don't worry if your answer is outside this, it is a very difficult  measurement (I got 30 Mpc when I tried it). How might you go about improving   your estimate?  By far the worst feature of this calculation is the assumption that these  galaxies out in a distant cluster are just like our own  galaxy. There are all sorts of reasons why this may not be true. For one,  this cluster is a lot bigger than our local group. Thus just by chance, you   might expect it to include a few rather large galaxies.   Also, nobody understands  how and why galaxies form, and it is quite possible that they form differently  in big clusters. So these galaxies may be nothing like our galaxy. How  can we get around this problem?  One way is to be selective in our choice of galaxies to measure. We know that  our galaxy, the Milky Way, is a spiral galaxy.   So try measuring a distance using only spiral galaxies in Hydra I;  see the appendix for a description of different galaxy types.  Has your answer changed? We also know that the Milky Way is the largest spiral  in our local galaxy cluster. So perhaps it is similar to the largest spiral  galaxies in Hydra I. Try getting a distance just from them.  Even this doesn't necessarily help us very much: we've no good reason  to believe that even large spiral galaxies in Hydra I should be like  the Milky Way. A much better assumption would be that the galaxies in  Hydra I are just like the galaxies in Virgo. After all, both clusters  are about the same size, mass and density. It seems very plausible  that the galaxies in Virgo should at least resemble those in Hydra 1.  Here is a better method (using Virgo) to try and measure the distance to  Hydra I:  \begin{enumerate}    \item Pick the ten biggest spirals and ten biggest ellipticals in each  cluster. Throw out any that look too bizarre \ -- \ you want to compare  the two clusters, so if in one you find, say, some seriously warped colliding  galaxies which aren't in the other cluster, you should throw them out. If  you can't find ten good ones, choose eight or five \ -- \ if you can find more  than ten, using them all will improve the accuracy of this method.    \item Measure the sizes of all these galaxies, using the eyepiece with  a scale. It doesn't actually matter how you do this, so long as you  do it the same way on each cluster. Try and think of a consistent way  of measuring the sizes.    \item Find the average apparent size (in arcmin) of the ten spirals and of the  ten ellipticals in each cluster, $r_{v}^{s}$ (mean size of Virgo  spirals), $r_{h}^{e}$ (mean size of Hydra ellipticals) etc.    \item For an object of a given size $r$ at a distance $d$, the angular  size will be $\theta = r/d$ where $\theta$ is measured in radians (why?).  This only applies if $d$ is very much greater than $r$ which is certainly  true here. Therefore, if we assume the ten brightest of each type of  galaxy are the same size in each cluster, the ratio of the distance to  Hydra I ($d_h$) to the distance to Virgo ($d_v$) is given by    \[ \frac{d_h}{d_v} = \frac{\theta_v^s}{\theta_h^s} \]    from the spirals, and    \[ \frac{d_h}{d_v} = \frac{\theta_v^e}{\theta_h^e} \]  from the ellipticals. These two numbers should be the same, to within the   errors. Are they? If you think one is more accurate, use it (justify  this decision in your notes). Otherwise, average the two numbers. It is  always worth measuring anything two ways to have a check.    \item The distance to Virgo is controversial, but most current estimates are  around 15 Mpc. Use this, plus the ratios measured above, to find the  distance to Hydra I.    \end{enumerate}  How does this answer compare with the earlier one? If they are different,  what is this telling us about the relative sizes of galaxies in Hydra I  and our galaxy? It could be that our distance to Virgo is wrong, as this   number is highly  controversial. Now that the Hubble Space Telescope has been fixed, an  international team (including several Australians) has been using it to   spot individual stars in Virgo galaxies, which will enable them to measure  the distance to Virgo accurately at last \ -- \ expect results in a year   or so.  \subsection{Hubble's Constant}  Reference: Zeilik, Gregory \& Smith, Introductory Astronomy \& Astrophysics,  Chapter 22.  The universe is expanding; every galaxy is moving away from every other  galaxy, and the further two galaxies are apart, the faster they are  receding from each other. This is known as Hubble's Law, after the  famous American astronomer who discovered the expansion of the Universe  back in the 1920's. It can be written as:  \begin{equation}  v = H_0 d  \label{eq:hubble}  \end{equation}  \noindent  where $v$ is the velocity at which something is moving away from us, $d$ its  distance from us, and the constant $H_0$ is known as Hubble's constant.  This constant is basically a measure of how fast the Universe is expanding.  If it is large, the universe is growing like crazy. We are going to use the  distance we measured to the Hydra I cluster to estimate $H_0$.  We have already measured the distance $d$ to Hydra I. Now we need the  velocity at which it is rushing away from us $v$. This is measured using  the Doppler effect. The gas in most Hydra I galaxies emits emission-lines  of Oxygen, Hydrogen and other elements. With a spectrometer on a telescope,  we can measure the Doppler shift in these lines, and hence the velocity $v$.  These velocities are listed in Table \ref{tab:hydragalvecolity}, for several of the brightest  galaxies in Hydra I.