Fixed bug in saving

Lab 2/Lab 2.ipynb

@@ -270,7 +270,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "Now we'll try a bit of astropy's machinery to deal with coordinates and observations.  We'll use the astropy [coordinates](https://docs.astropy.org/en/stable/coordinates/index.html)coordinates, [time](https://docs.astropy.org/en/stable/time/index.html) and [units](https://docs.astropy.org/en/stable/units/index.html) packages. Check out their documentation pages for some examples on how to use them. **To use them, we first need to import the relevant bits, so put these lines in a notebook cell:**\n",
+    "Now we'll try a bit of astropy's machinery to deal with coordinates and observations.  We'll use the astropy [coordinates](https://docs.astropy.org/en/stable/coordinates/index.html), [time](https://docs.astropy.org/en/stable/time/index.html) and [units](https://docs.astropy.org/en/stable/units/index.html) packages. Check out their documentation pages for some examples on how to use them. **To use them, we first need to import the relevant bits, so put these lines in a notebook cell:**\n",
     "```\n",
     "from astropy import coordinates, time\n",
     "from astropy import units as u\n",
@@ -609,6 +609,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
+    "In the ds9  ''Zoom'' menu bar (not the Zoom button), click ''Invert Y'' to flip the image upside down. This will facilitate comparison with other images that are in a more standard format. (Notice, for future reference, that other flip and rotate operations are possible.)\n",
     "After doing a Y-invert (see below), your image should look like this, except for the little circles and numbers, and the color scheme I used to make the numbers show up better.\n"
    ]
   },
@@ -616,7 +617,161 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "![test](M67_circled.jpeg)"
+    "![test](./M67_circled.jpeg)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "This is an image of the Messier 67 [open cluster](https://en.wikipedia.org/wiki/Open_cluster). Take some time to google \"Messier Catalog\" and browse some of the beautiful images of these different objects. **What is your groups favourite object?**"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "*Your answer here*"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Now open a browser and go to [http://www.sky-map.org](http://www.sky-map.org). This is a nice piece of planetarium software (similar to the Stellarium page we used earlier). Go to ''Home,'' and enter the name of the cluster in the ''Find Object'' window. When the map comes up, point to the small elliptical icon in the upper left called ``DSS,'' and\n",
+    "select ``DSS2 All Sky Survey.'' These data come from various releases of the [Digital Sky Survey](https://irsa.ipac.caltech.edu/data/DSS/). Zoom in 3 or 4 clicks on the size scale, and you should see a familiar star cluster (a bit off center). Drag it\n",
+    "to the center of the window, and zoom it to whatever degree makes you comfortable. Now when you drag the cursor over a star image you\n",
+    "will see lots of information about each star, including a long catalog number, and (most importantly) the star’s equatorial coordinates."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Identify the 4 numbered stars in the image above on the ```sky-map.org``` site, and list their RA, $\\delta$ values in the table below, 1 row per star. Also measure the $\\{x,y\\}$ coordinates\n",
+    "of each star on ```cluster.fits```, using the cursor to pick out the brightest point in each star. Do this carefully, zooming so that setting the cursor is\n",
+    "easy, and adjusting the ``scale'' options so you can easily see the brightness variations inside the star images.  Make a subjective guess about the error (in pixel units) with which you can measure the star positions. Put this in the table too, under error\\_g."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Star| RA | $\\delta$ | $x$ position | $y$ position | error_g\n",
+    "---|---|---|---|---|---\n",
+    "1 |---|---|---|---|--- \n",
+    "2 |---|---|---|---|--- \n",
+    "3 |---|---|---|---|--- \n",
+    "4 |---|---|---|---|--- "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "What do you think is your largest source of error (the one that dominates your estimate of\n",
+    "error g)?"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "*Your answer here*"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Now compute the distances between various pairs of stars, as given below. Do this first by using the difference in RA and d that you obtained\n",
+    "from ```sky-map.org```. Formally this is an exercise in spherical trigonometry, but because all of these stars are very close together on the sky, we\n",
+    "may use small-angle approximations. In this case we get sufficient accuracy by taking}\n",
+    "$$\n",
+    "\\Delta r = \\sqrt{(\\Delta \\delta)^2 + \\left(\\Delta {\\rm RA} \\cos \\delta \\right)^2},\n",
+    "$$\n",
+    "where $\\Delta r$ is the angular separation between two stars, $\\Delta \\delta$ is the separation in Declination, and $\\Delta {\\rm RA}$ is the separation in Right Ascension, with\n",
+    "all angles are expressed in units of angle (use arcsec). Remember that RA is normally expressed in units of time, not angle -- one second of\n",
+    "RA (the difference between RA = 08:30:00 and 08:30:01, for instance) equals 15 arcsec. You should think about where the factor $\\cos(\\delta)$ comes\n",
+    "from. Try computing the RA and Dec values as arcseconds first by hand, and then you can double check using astropy coordinates. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "#Your code here"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "*Your answer here*"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "**Also compute the separation between these pairs of stars in units of pixels, using your measured values of $x$ position and $y$ position. In this case the\n",
+    "normal Pythagorean law may be used, with no $\\cos(\\delta)$ factor. (Think about why.) Use your estimates of error\\_g and standard propagation-of-error\n",
+    "rules (see the textbook by ***Taylor*** linked on the Lab 2 tab in Gauchospace for a refresher) to estimate the errors in these separations which we will call error $_p$. In the space below, show the formula(s) you used for calculating the error $_p$ values. Then put all of the data into the table below. Expand the number of rows as necessary.**"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Star Pair | $\\Delta r$ (arcsec)| $\\Delta p$ (pixel) | Error $_p$ (pixel)| Scale ($\\frac{arcsec}{pixel}$) | Error $_{s}$ ($\\frac{arcsec}{pixel}$)\n",
+    "---|---|---|---|---|---\n",
+    "(1,3) |---|---|---|---|--- \n",
+    "(1,4) |---|---|---|---|--- \n",
+    "(2,3) |---|---|---|---|--- \n",
+    "(2,4) |---|---|---|---|--- \n",
+    "(3,4) |---|---|---|---|--- "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "**For each star pair, compute the image scale $\\Delta r/\\Delta p$ in units of arcsec/pixel, and enter this value in the table. Use Taylor’s error propagation rules, starting from your estimates of error $_p$, to estimate the error in your derived value for the image scale (which we will call error $_s$).\n",
+    "Assume that the star separations derived from ```www.sky-map.org``` positions have negligible errors. Put your error $_s$ values in the table.**\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Now you have five not-quite-independent measurements of the image scale. Do they agree with each other, within the plausible errors?  If not (and especially if the disagreement is very large), the most likely explanations are (a) there is a mis-identified star, or (b) there is an error in computation. In either of these cases, you should go back and correct the error before proceeding.\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "**Average your results for image scale (we will do weighted averages later), and estimate the uncertainty of this mean value, again using the rules\n",
+    "described in Taylor’s book. From the image scale, compute the telescope focal length and its uncertainty. Compare this to the focal length\n",
+    "value found in the fits image header. Show your work.**"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "#Your code here"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "*You answer here*"
    ]
   },
   {