Astronomy 15 - Homework 1

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Theodora Wilkins
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1 Astronomy 15 - Homework 1 Here are some exercises to get you used to celestial coordinates and calculations. 1) For an observer at latitude l, the altitude a of an object at hour angle H and declination δ is given by a = arcsin(cos l cos H cos δ + sin l sin δ), where atmospheric refraction has been ignored. Clearly, when the altitude is zero (on the horizon), the expression in parentheses is also zero. Hanover, NH is at l = 43.7 degrees. Compute the hour angle of the setting sun on the winter solstice; how long is the sun above the horizon that day? Repeat for the summer solstice. 2) Here s a way of computing the angle between two objects in the sky. (a) First, you transform the celestial position to Cartesian unit vectors. Show that a unit vector to an object with celestial coordinates (α, δ) has components z = sin δ, x = cos δ cos α, and y = cos δ sin α where the x-axis points toward (α, δ) = (0, 0), the y-axis points toward (α, δ) = (6 h, 0), and the z-axis points toward the north celestial pole. (Hint: How long is the projection of the vector onto the x-y plane?) Verify explicitly that the resulting vector has length = 1 (that s the easy part). (b) Now recall that the inner or dot product of two vectors a = (x a, y a, z a ) and b = (x b, y b, z b ) is a b = x a x b + y a y b + z a z b = ab cos θ, where a and b are the magnitudes of the two vectors and θ is the angle between them. Use this to calculate the angle between a star at α = 0 h, δ = 0 and a star at α = 6 h, δ = 30. Remember, 1 h = 15. Interlude: Linux on the Astro Lab Machines Much of the astronomical community uses Unix or Linux as the computer operating system of choice, instead of Microsoft Windows or the various Macintosh systems. Unix was first developed back in the 1970s at Bell Labs. Back then, computers were big expensive things and most of them had many users at the same time, so it was important not to let any one user get control of the machine, deliberately or inadvertently. Therefore, stability was engineered into the system from the start each user is allowed to see and manipulate only certain files, and in principle a user s program cannot crash the whole system (in practice this can happen, but very rarely). By contrast, the PC operating systems were developed around the idea that a single user controls the system. Features like security and stability were afterthoughts if the thing crashed, you just got the blue 1

2 screen of death, or the little bomb icon, and a three-fingered salute (ctrl-alt-delete) would get you out. Linux was the brainchild of a Finno-Swedish programmer, Linus Torvalds. Seeing the proliferation of inexpensive PC hardware, he wrote a little system which acted like Unix, put it out over the Internet, and invited others to contribute functionality. In a remarkably short time, the volunteer efforts of thousands of enthusiasts worldwide created a robust, powerful, full-featured operating system which is mostly very similar to Unix. Linux is free but creating a working Linux system yourself from the resources out on the web is painfully difficult. Accordingly a number of companies have sprung up which package up Linux software in user-friendly bundles for a modest price. RedHat Linux is the most popular flavor in the U.S. market. Nearly all software for Linux is free. That s right, it costs nothing. Some specialized stuff does cost money, but there s often a freeware alternative for many tasks. We have a suite of machines in Room 200 which are dual-boot, that is, they run either Linux or Windows. By default they come up in Linux. Each of you will get a user account for these machines. This is necessary because all Linux or Unix machines require them you can t just turn it on and start typing. The advantage, as noted above, is that they don t (or hardly ever) crash. You user account will come with a password. You should immediately change your default password to something secure which you can remember. Hackers routinely try to break into machines by cracking passwords, so it s important to make your password uncrackable. The rules are: Don t use any proper name or English or other language word Don t use something obvious like an encoded birthdate Make your password reasonably long (like 8 characters) Mix upper- and lowercase letters, numbers, and punctuation marks Ideally, come up with some unforgettable mnemonic for your password. Once you ve thought up a password, sit down at one of the machines. If it s not on, turn it on with the button it will take a minute or so to go through the boot sequence. When it comes up with a window which prompts for a username, type it in, and then type your default password to log on. Then the machine will come up in the X windows display. X is a window manager which runs under Linux - Linux itself will run from a command line alone, but that s somewhat limiting. The particular flavor of X you re running is called fvwm2. By default there will be one window an X terminal which will be ready to accept text. Put the mouse in the window, and change your password by typing yppasswd and answering the prompts. You ll be asked to type your new password twice to avoid typing errors. 2

3 You might have to select the text window by clicking on it. Nearly everything in Linux is done with the left mouse button, which is the one under your index finger if you re right-handed. In the instructions below, the left button is understood unless otherwise indicated. You can move the text window around by clicking on the top bar and dragging it, and you can resize it by clicking accurately on the corner of the border. Try it, it ll be helpful later. You re ready to go! Exercises with xephem Xephem is a very nice desktop-planetarium calculator package written by Elwood Downey. He has a free version available on the Web, and for something like $80 you can get a CD- ROM version with lots of extra goodies *. We just need the free version, which is installed on these machines. To start xephem, place the mouse in the terminal window (you may have to click on the border with the left mouse button to activate it for input), and type xephem &. The ampersand launches the job in background and leaves the terminal free for other input. A control panel for xephem will come up after a few seconds. X1: Let s repeat the sidereal/solar demonstration we did in class. Look at the lower right portion of the control panel, which says Looping. Click on N Steps and select ; click on Step and select 24:00:00 ; click on Pause, and in the dialog box (which you have to click on to get access to) put in 1 second. Now, go to menus at the top of the window and under View select Sky view.... This will pop a window showing the sky. Finally, at the bottom of the box, click on the Update control bar. The display will start to move. Watch it for a while. Write a few sentences to describe what you see. Return to the control panel and click on the bottom bar, which is now labeled Stop. Return to the Step item and select Sidereal Day. Start the demo again with Update. Again, watch it for a while. Write a few sentences to describe what you see. X2: Let s trace the sun through the year, in an interesting way. Select a local time sometime when the sun is up around local noon works well. You can experiment with the UTC time to do this just change it and hit the update button (but be sure to change the number of steps back to 1 ). Go to the Looping items and set it up to have a step of 7 days, and to run 52 steps. Under the View menu, select Data Table. * This is available from clearskyinstitute.com. If you re into this stuff it could be a good investment, esp. since the free version is quite difficult to install properly 3

4 Under the tools menu, choose Plot values. This will bring up a dialogue box. Click create plot file. Under the button labeled Browse... is a box for entering a file name. Click in this box, erase what s there, and enter something like /net/myuser/myplot.plt (don t type this! Read on for what to really type). This is the name of the file you ll be sending output to; the format is a first example of a path name. In this example /net/myuser is the name of the directory you re writing to, the words separated by slashes are directory levels (so /net is the top directory, myuser is the name of the subdirectory, and so on. To figure out what to actually give for your directory name, go back to the text window from which you launched xephem and type pwd (which stands for print working directory ), which will print out the name of your own personal directory (your own top directory). Figure out your own path from the response. Now click on the select fields radio button. Boxes will appear around all the items in the data table. In the row labeled Sun, click on Az and Alt. This will be where the sun appears in the sky on each date. The fields Sun.Az and Sun.Alt will appear under the X and Y columns in the plot control window. Click again on Create plot file and don t worry about the append/overwrite choice box which comes up just click either one (but overwrite if it s your second time through). Time to run it. Click on Update again, and watch the sky go week-by-week through the year. When it s done, click close on the Plot Control window, then select it again (under tools) not sure why one has to close it and open it, but it looks as if one must and then Show plot file. Describe what you see. In the Control menu under the plot, there s a Print item, which might work. Try it and see! X3: Spend a little time playing around with the program and its controls. When you move the mouse around on the skyview display, it shows the coordinates of the mouse pointer in various systems look at these and make sure you understand what you re looking at. Under Control in the sky view display you can do lots of things, like turn on and off the boundaries between the constellations (the constellations have official boundaries, a relatively modern invention, so that any point in the sky belongs to a well-defined constellation, unless it s exactly on a border). If you point at an object and click on the right mouse button, you can get a little synopsis about what the object is. The little buttons along the side of the sky view display control options affecting the appearance of the display. You can enlarge the sky view by resizing the window. You can get a nice picture of the solar system as well under the View menu. Watch the moons of Jupiter for a while there are lots of possibilities. Exercises with Skycalc I wrote a little program which does time-and-the-sky calculations, called skycalc. doesn t do nice fancy displays like xephem, but it s simple to use. It Go back to the text window, and type skycalc. It will ask you for what site you re observing from; it s s for Shattuck (followed by the enter key, nothing happens til you type enter). Now you can use the program to check some of the answers to your earlier work. 4

5 a) You can enter a date with y or whatever this means 2002 April 5, and type a to get an almanac for that date. The output on the screen will include sunrise and sunset times which you can use to check the length-of-day calculation. b) Here s a neat trick; enter your date of birth with y, and the time with t (if you were born at 3 minutes past 7 in the evening, for example). If you hit = you will get a big output which includes the Julian date, a big number over 2 million. Write it down. Now set the date to now, which you can do easily by typing x T 0, and hit = again. Write down the Julian date again. The Julian date is a timekeeping system which is simply a sequential day number, with a fractional part to deal with time of day. Julian date zero was back in 4713 BCE, so all historical dates are positive (and the numbers are huge). Now you can easily compute your age in days, simply by subtracting the two Julian dates this bypasses all the complexities of the calendar for any kind of timekeeping. All you need is a routine which reliably converts back and forth from Julian date. c) As we saw in class, the earth s axis precesses gradually with time, which leads to a slow change in the coordinates in an object (even if the object isn t moving). Let s see how much of a difference this makes. There s an interesting star called SS433, which is at 1950 coordinates α = 19 h 09 m 21 s.3 and δ = Suppose you were to mistakenly aim the telescope at the 1950 coordinates at the present time. How far off would you be? To answer this, type e 1950 in skycalc, which tells the program to expect 1950 coordinates; then enter the ra and dec with r and d , and finally hit = to see the result. Write down the present coordinates. How far off is this? You can figure this out by considering the sky locally to be like a little piece of flat graph paper and using the Pythagorean theorem to compute the distance. Complications include (1) converting to decimal units before subtraction; (2) multiplying the RA difference by 15 to change it to degrees; and (3) multiplying the RA difference by cos δ to take account of how the lines of constant RA crowd together as you go off the equator. Convert your answer to minutes of arc (60ths of a degree), and compare it to the typical 5 arcmin finder field of a large telescope. 5

AY120A,B CHEAT-SHEET FOR SPHERICAL COORDINATE TRANSFORMATION

AY120A,B CHEAT-SHEET FOR SPHERICAL COORDINATE TRANSFORMATION Carl Heiles In our never-ending attempt to make your life easier, we present you with the quickest of quick summaries of spherical coordinate