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\subsection{Aim}  To find lots of uncatalogued asteroids. To work out where the asteroid  belt is, and roughly how thick it is. To put a lower limit on the number of  asteroids in the asteroid belt. You should also learn a lot about measuring  photographic plates.  \subsection{Background}  Most large professional telescopes look at really tiny areas of sky in  immense detail. This is great for studying a particular planet or galaxy,  but for certain problems is rather bad. One such problem is finding  asteroids. Contrary to the impressions given by various science-fiction movies, asteroids are quite spread out, so you could spend years  waiting for an asteroid to blunder past the particular tiny bit of  sky you are staring at.  Luckily, some clever telescope designers have come up with a solution. They  devised a special type of telescope called a {\bf Schmidt Telescope}. This  has specially designed optics to take pictures of very wide areas of sky,  allowing you to look for rare or spread-out objects.  You will be using data taken with the best Schmidt telescope in the world,  which is jointly run by the UK and Australia, and is sited at Siding  Spring Observatory, near Coonabarabran in NSW. Since it was completed in  1973, this telescope has been systematically mapping the sky. Each picture  it takes covers about $6^{\circ} \times 6^{\circ}$ (for reference, the  diameter of the moon is about $0.5^{\circ}$). In collaboration with a  northern telescope (at Mt Palomar in California) the whole sky has now been  photographed. The photographs are called plates, as they are thin glass  sheets covered on one side with photographic emulsion.  In the lab, we have a collection of film copies of Schmidt plates. You  should browse through some of them \ -- \ there are some amazing pictures  of galaxies, nebulae, comets etc. See how much data is on each picture;  there can be as many as a million stars recorded on one picture  of a bit of the Milky Way (nearly all those little dots really are stars, not  dirt!). Note that these film copies are research grade material, not  special teaching stuff \ -- \ a lot of professional astronomers spend their  lives poring over these plates trying to find quasars, galaxies, comets,  the tenth planet, sputniks and so on.  {\bf Important number: the plate scale of pictures taken with this telescope  is 1.12 arcmin per mm.}  \subsection{Measurements}  You will be using plate number J2137 (also known by the coordinates of its  centre on the sky, 12:27+13:30). This plate was taken  for a study of a giant cluster of galaxies, the Virgo Cluster, which you  can easily see in the middle. This picture was taken with a 70 minute  exposure, and objects as faint as about 22nd magnitude can be seen (if  you try real hard). South is at the bottom of the picture, and East is to  the left. Positions on the plate should be quoted as the number of mm  right and up from the bottom left-hand (SE) corner.  This plate was taken with the telescope tracking \ -- \ ie following the  stars as they move across the sky. This means that stars, galaxies, quasars  etc appear as nice sharp images on the plate. However, if something up in  space moved appreciably during the exposure, it will appear not as a dot but  as a line.  This is how you hunt for asteroids; you search for things on the plate which  look like short lines - about 1 mm long. Put the film copy of the plate on  the light table. Use a magnifying lens and see  if you can find any (hint - there are several dozen on the plate). If you have  trouble, ask your demonstrator to point a few out to you. These are not known  asteroids \ -- \ only one of the asteroids on this plate is in any catalog  (asteroid Suleika, 230mm right and 212mm up from the bottom left-hand  corner). So these asteroids you are finding have no names, not even numbers,  and nobody knows anything about them.  Once you've found an asteroid, make a note of where it is so you can  find it again. The best way to do this is with an overhead projector  transparency and a washable marker pen. Please don't write directly on  the plate, or write on the overhead sheet with a pen or anything hard  as you will damage it. To get your overhead projector sheets aligned right,  mark a few really bright stars or galaxies on them.  As you will soon find out, there is a lot of crud on the light table and the  film covers. Often things you think might be asteroids will turn out to  be dirt - try moving the plate around on the light table and brushing both  sides before you decide an object is an asteroid. Other things which may look  like asteroids at first are edge-on spiral galaxies (there are some beauties  in the Virgo cluster) and random groups of stars that happen to line up.  With practice you will soon be spotting the real asteroids.  \subsection{Where are the Asteroids?}  The first exercise with the plate is to work out where these asteroids are  in the solar system. The coordinates of the centre of the plate are  12 hours 27 min in right ascension, 13 degrees and 30 minutes in declination  (northern hemisphere). See where this is on your planispheres, and on  the sky globe in class (now you know why it is called the Virgo cluster).  Is it near the ecliptic plane? At what angle does the ecliptic plane  lie, compared to the plate boundaries?  The plate was taken on 28th March 1976. Using the Skyglobe program on  the PCs, work out where the sun was at on this date (its right ascension  and declination). Compare the position of the Sun and of the plate on  a globe of the sky. You will see they are almost at opposite locations.  Measure the direction of motion of a few asteroids, using a protractor.  Are they moving along the Ecliptic? Most of them will be; these are the  asteroids in the asteroid belt, and like most of the planets they are  orbiting the Sun in the Ecliptic plane. You may however find a few  maverick asteroids that are moving with quite different speeds or  directions. These rogue asteroids are most likely nearer the Earth on  highly eccentric orbits. One much like this may well have been responsible  for eradicating the dinosaurs 60 million years ago. Take careful note  of where they are and tell your demonstrator \ -- \ the class should build  up a collection of these.  So, we now know that most of these asteroids are moving along the  ecliptic, and as they are in the opposite direction from the Sun, they must  be further out that the Earth. Can we work out how far away they are?  The answer is yes, but only if we make a couple of key assumptions.  Firstly, we must assume that they are in roughly circular orbits \ -- \ not  too eccentric. This is true of all known belt asteroids, though not  of the maverick near-Earth ones. We will also assume that they are orbiting  along the plane of the Ecliptic, in the same direction as the Earth (and  all the other planets). To simplify the maths, we assume that they are  exactly opposite the Sun from us and that both the asteroids and the Earth  are moving along the Ecliptic in perfectly circular orbits.  The motion of the asteroid as seen from the Earth will have two causes.  Firstly, the asteroids will be moving slowly around their orbits. Secondly,  the Earth is speeding around its orbit, so from the Earth, the asteroids  will tend to move backwards. As the Earth is moving faster than the asteroids,  this latter effect is the biggest one, so they will appear to move backwards.  Measure how fast they are moving. You do this by measuring the length of the  little line on the plate. As you know the plate scale (1.12 arcmin per mm)  and the exposure time (70 minutes), you can work out the angular speed  at which the images move. You should do this for a few asteroids \ -- \  the more you measure, the more accurate your answer will be when you  average them all.  Now, using Newton's law of gravity, you can work out how fast the Earth  is moving round the Sun (you will need to know the mass of the Sun,  $ 2.0 \times 10^{30}$ kg, and the distance from the Earth to the Sun,  $1.5 \times 10^{11}$ m). If the asteroid were a distance $r$ from the  Sun, how fast would it be moving? And what would its backward angular speed  be when viewed from the Earth? Using the observed backward angular speed,  solve the equation and Bingo! You have $r$, the distance of the asteroids  from the Sun (The equation in a cubic, and so not trivial to solve.  However, we don't need a precise solution, s you can solve it numerically  or just plot it as a graph).  So where are they? You now know how far their orbits are from the Sun,  compare this with the orbits of the planets. Which planets will they pass  closest to? Anything unusual about these planets which might explain  why the asteroids are near them?  \subsection{The Asteroid Belt}  In the first exercise, you discovered that the asteroids were orbiting the  Sun, and measured their distance from the Sun. Now, we will work out how  thick the asteroid belt is. To do this, you should try and measure  every asteroid on the plate. Search for them systematically; figure out  some methodical way of working your way across the plate to make sure you  don't miss any. The really faint ones will always be hard to find, so you  might want to keep a separate list of possible asteroids which you are  not sure of.  The first thing to check, once you've measured positions and angular  speeds for all the asteroids, it whether they are concentrated near the  ecliptic plane. You have already worked out where the plate lies  compared to this plane. Divide the plate in half; the half nearest and  furthest from the Ecliptic. Count the number of asteroids per unit area  in each half. Are there more near the plane? Further from it? Is the  number constant? What does this tell you about the shape of the asteroid  belt?  The next thing to check is the spread in distances from the Sun. For  each asteroid, measure its angular speed, and hence its distance from the  Sun. Try plotting a histogram of these values. How spread out are they? Are there any  asteroids well outside the normal belt?  As you will be finding out, spotting things on these plates is no slouch.  You might want to think about the following biases and how to avoid them.  A lot of your marks will depend on how clever you are at this. But  don't feel alone and victimised by these problems; a lot of professional  astronomers (including the author) have spent a lot of their lives worrying  about these biases. It really is the mark of a good astrophysicist to  know what is good and bad about your data, to anticipate things that  could go wrong and figure out ways of solving them.  \begin{itemize}    \item If you just wander around the plate spotting things, you will  inevitably pay more attention to some areas than others. Be systematic!    \item As you get tired and bored, you will probably start missing asteroids.  If you started from the top of the plate and worked down (for example),  this might mean that you found a whole lot more asteroids at the top than  at the bottom. This might mean you see numbers of asteroids increasing  towards or away from the Ecliptic where no such effect really existed. A  possible way out it to search the plate in a more random way than just  starting at one end and moving through.    \item The opposite of the above; as time goes on you get more experienced  and spot more asteroids.    \item You don't have to measure the whole plate yourself; share your  results. But this introduces a new bias. Not everyone will be equally  thorough in finding asteroids. You should cross check results. Perhaps  you should all do a few overlapping regions, and check that you find the  same number of asteroids.    \item Fainter asteroids are harder to spot than bright ones, and more  easily confused with dust. Make sure you aren't ignoring faint ones or  bright ones; they might be in different parts of the belt.    \item Short and long asteroid trails may be easier or harder to find. If you  are biased in length, what effect will this have on the histogram of  distances from the Sun you are measuring?    \end{itemize}  \subsection{Advanced Exercises}  You have measured the radial thickness of the asteroid belt. Using  your counts of asteroids per square degree, work out the density  (asteroids per square km) of the belt. How does this compare with  asteroid belts in Sci-Fi movies? If each asteroid has a diameter of  1km and is a perfect sphere, what are the odds of a rocket hitting an  asteroid as it shoots through the belt at high speed? Note that this  is a lower limit; there could be lots of small dark asteroids too faint  to see on this plate.  The apparent magnitude of the asteroid Suleika (230 mm right and 212 up from  the bottom left-hand  corner) can be measured from this plate using a microdensitometer.  It is 13.3. Using this, combined with the distance of Suleika from the  Sun (which you can measure from its motion), you should be able to work  out how big Suleika is. You will need to assume it is a sphere, and that  it reflects 6\% of the light that falls on it. It will be helpful to know  that the apparent magnitude of the Sun is -26.1. If the density of Suleika is  typical of meteor rock ($3.5 \times 10^3$) what is the mass of Suleika? If all  asteroids have the same mass, what is the density of the asteroid belt?  Estimate a thickness for the asteroid belt (how high it  extends above and below the ecliptic), and work out the total mass of the  asteroid belt. How does this compare with the mass of a planet? Is this  good news for the theory that the asteroids were formed when a planet  exploded?