<html><head><title>Addendum 4 Optics, Telescope</title><style type="text/css"> p { margin: 0 0 0 0; font-family: Arial, sans-serif; font-size:10pt; font-weight: normal; font-style: normal; vertical-align: baseline; color: black; text-decoration: none; } div { border-width:0px; border-style: solid; border-color: red; } body { background-color: #fff; } p img { vertical-align: middle; } p b, li b { font-weight : bold; } p i, li i { font-style : italic; } p u, li u { text-decoration : underline; } p so, li so { text-decoration : line-through; } p sub, p sub { font-size : 70%; vertical-align: sub; } p sup, p sup { font-size : 70%; vertical-align: super; } ul { padding-left : 2em; /* for mozilla list marker box */ margin-left : 0; /* for IE list marker box */ } ol { padding-left : 2em; /* for mozilla list marker box */ margin-right : -0.5em; margin-left : 0; /* for IE list marker box */ } p.Style1, li.Style1 /*Normal*/ { margin-left: 0pt; margin-right: 0pt; text-align: left; font-family: 'Arial', sans-serif;; 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top: 72.75pt; left: 30pt; width: 282.5pt; "><a name="" /><p class="Style1"><span style="font-size: 12pt; font-family: 'Arial', sans-serif;"><b><span>Addendum 4: Optics, Telescopes, Angular Sizes </span></b></span></p></div><div id="mc-region-19" style="position: absolute; top: 110.25pt; left: 24pt; width: 408.5pt; "><a name="" /><p class="Style1">1. <b>Magnification of refracting telescope</b><span style="font-family: 'Arial', sans-serif;"><b><i><span>. </span></i></b></span><span style="font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;">T</span></span><span style="font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;">he magnification of a refracting telescope is given by the ratio of the focal length of the objective lens (the large one in front) to the focal length of the eyepiece. </span></span></p></div><div id="mc-region-398" style="position: absolute; top: 168pt; left: 216pt; width: 178.5pt; "><a name="" /><img border="0" src=".\addendum 4 optics telescopes_images/IMG0004_273747625.PNG" id="generatedImage398" path=".\addendum 4 optics telescopes_images/IMG0004_273747625.PNG" style="width: 178.5pt;height: 129.75pt;"></img></div><div id="mc-region-403" style="position: absolute; top: 164.25pt; left: 36pt; width: 48.5pt; "><a name="" /><p class="Style1"><i>Example: </i></p></div><div id="mc-region-404" style="position: absolute; top: 165pt; left: 96pt; width: 36pt; "><a name="" /><img border="0" src=".\addendum 4 optics telescopes_images/IMG0005_273747625.PNG" id="generatedImage404" path=".\addendum 4 optics telescopes_images/IMG0005_273747625.PNG" style="width: 36pt;height: 16.5pt;"></img></div><div id="mc-region-405" style="position: absolute; top: 165pt; left: 156pt; width: 46.5pt; "><a name="" /><img border="0" src=".\addendum 4 optics telescopes_images/IMG0006_273747625.PNG" id="generatedImage405" path=".\addendum 4 optics telescopes_images/IMG0006_273747625.PNG" style="width: 46.5pt;height: 16.5pt;"></img></div><div id="mc-region-406" style="position: absolute; top: 196.5pt; left: 54pt; width: 33.75pt; "><a name="" /><img border="0" src=".\addendum 4 optics telescopes_images/IMG0007_273747625.PNG" id="generatedImage406" path=".\addendum 4 optics telescopes_images/IMG0007_273747625.PNG" style="width: 33.75pt;height: 35.25pt;"></img></div><div id="mc-region-408" style="position: absolute; top: 207pt; left: 114pt; width: 47.25pt; background-color: inherit; "><a name="" /><img border="0" src=".\addendum 4 optics telescopes_images/IMG0008_273747625.PNG" id="generatedImage408" path=".\addendum 4 optics telescopes_images/IMG0008_273747625.PNG" style="width: 47.25pt;height: 12.75pt;"></img></div><div id="mc-region-410" style="position: absolute; top: 302.25pt; left: 36pt; width: 407.75pt; "><a name="" /><p class="Style1">2. <b>Light gathering power</b>. The light gathering power of a telescope, which is the telescope's most important parameter, is simply the effective collecting area. This is normally the area of the primary mirror or the objective lens.</p></div><div id="mc-region-412" style="position: absolute; top: 349.5pt; left: 78pt; width: 37.5pt; "><a name="" /><img border="0" src=".\addendum 4 optics telescopes_images/IMG0009_273747625.PNG" id="generatedImage412" path=".\addendum 4 optics telescopes_images/IMG0009_273747625.PNG" style="width: 37.5pt;height: 28.5pt;"></img></div><div id="mc-region-414" style="position: absolute; top: 356.25pt; left: 138pt; width: 172.25pt; "><a name="" /><p class="Style1">where <i>d</i> is the diameter (mirror or lens)</p></div><div id="mc-region-472" style="position: absolute; top: 380.25pt; left: 30pt; width: 390.5pt; "><a name="" /><p class="Style1"><i>Example: The Hubble telescope has a primary mirror d = 2.4m, while the Iowa Robotic Telescope's (Rigel) primary mirror is d = 37cm. What is the ratio of light gathering powers of HST to Rigel?</i></p></div><div id="mc-region-469" style="position: absolute; top: 422.25pt; left: 108pt; width: 62.25pt; "><a name="" /><img border="0" src=".\addendum 4 optics telescopes_images/IMG0010_273747640.PNG" id="generatedImage469" path=".\addendum 4 optics telescopes_images/IMG0010_273747640.PNG" style="width: 62.25pt;height: 33.75pt;"></img></div><div id="mc-region-470" style="position: absolute; top: 435pt; left: 192pt; width: 44.25pt; background-color: inherit; "><a name="" /><img border="0" src=".\addendum 4 optics telescopes_images/IMG0011_273747640.PNG" id="generatedImage470" path=".\addendum 4 optics telescopes_images/IMG0011_273747640.PNG" style="width: 44.25pt;height: 12.75pt;"></img></div><div id="mc-region-421" style="position: absolute; top: 464.25pt; left: 24pt; width: 408.5pt; "><a name="" /><p class="Style1" style="margin-left: 0pt; margin-right: 0pt; text-align: left;">3. <b>Angular Resolution</b><span style="font-family: 'Arial', sans-serif;"><b><i><span>.</span></i></b></span><span style="font-family: 'Arial', sans-serif;"><i><span style="font-weight: normal;"> A telescope's angular resolution is a measure of the ability to resolve fine detail</span></i></span><span style="font-family: 'Arial', sans-serif;"><span style="font-weight: normal;">. For a telescope with diameter </span></span><span style="font-family: 'Arial', sans-serif;"><i><span style="font-weight: normal;">d</span></i></span><span style="font-weight: normal;"><span style="font-family: 'Arial', sans-serif;"> or an interferometer with separation </span></span><span style="font-family: 'Arial', sans-serif;"><i><span style="font-weight: normal;">d</span></i></span><span style="font-family: 'Arial', sans-serif;"><span style="font-weight: normal;">, it is given by:</span></span></p></div><div id="mc-region-422" style="position: absolute; top: 499.5pt; left: 180pt; width: 26.25pt; "><a name="" /><img border="0" src=".\addendum 4 optics telescopes_images/IMG0012_273747640.PNG" id="generatedImage422" path=".\addendum 4 optics telescopes_images/IMG0012_273747640.PNG" style="width: 26.25pt;height: 28.5pt;"></img></div><div id="mc-region-518" style="position: absolute; top: 536.25pt; left: 30pt; width: 390.5pt; "><a name="" /><p class="Style1" style="margin-left: 0pt; margin-right: 0pt; text-align: left;">where <i>d</i> is the diameter (or separation for an interferomter), <span style="font-size: 10pt; font-family: 'Symbol', sans-serif;"><span style="font-weight: normal; font-style: normal;">l</span></span><span style="font-size: 10pt; font-family: 'Symbol', sans-serif;"><span style="font-weight: normal; font-style: normal;"> </span></span><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;">is the wavelength, and </span></span>the resolving angle <span style="font-size: 10pt; font-family: 'Symbol', sans-serif;"><span style="font-weight: normal; font-style: normal;">q</span></span><span style="font-size: 10pt; font-family: 'Symbol', sans-serif;"><span style="font-weight: normal; font-style: normal;"> </span></span><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;">is in radians. Note: </span></span><span style="font-weight: normal; font-style: normal;"><span style="font-size: 10pt; font-family: 'Symbol', sans-serif;">l</span></span><span style="font-size: 10pt; font-family: 'Symbol', sans-serif;"><span style="font-weight: normal; font-style: normal;"> </span></span><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;">and d must be in the same dimensional units (e.g. meters). </span></span><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;">To convert radians to arcseconds (a much more sensible unit for angular resolution, multiply by 360*3600/2*</span></span><span style="font-size: 10pt; font-family: 'Symbol', sans-serif;"><span style="font-weight: normal; font-style: normal;">p</span></span><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;"> = 206,265. </span></span></p></div><div id="mc-region-430" style="position: absolute; top: 626.25pt; left: 30pt; width: 385.25pt; "><a name="" /><p class="Style1"><i>Example: What is the angular resolution of the VLA radio telescope array in New Mexico at an observing frequency of 43 GHz? (1 GHz = 10</i><i><sup>9</sup></i><i> Hz). The maximum baseline is 30 km. </i></p></div><div id="mc-region-432" style="position: absolute; top: 674.25pt; left: 36pt; width: 192.5pt; "><a name="" /><p class="Style1">First convert from frequency to wavelength: </p></div><div id="mc-region-437" style="position: absolute; top: 669pt; left: 234pt; width: 55.5pt; "><a name="" /><img border="0" src=".\addendum 4 optics telescopes_images/IMG0013_273747640.PNG" id="generatedImage437" path=".\addendum 4 optics telescopes_images/IMG0013_273747640.PNG" style="width: 55.5pt;height: 18.75pt;"></img></div><div id="mc-region-441" style="position: absolute; top: 668.25pt; left: 312pt; width: 26.25pt; "><a name="" /><img border="0" src=".\addendum 4 optics telescopes_images/IMG0014_273747640.PNG" id="generatedImage441" path=".\addendum 4 optics telescopes_images/IMG0014_273747640.PNG" style="width: 26.25pt;height: 27.75pt;"></img></div><div id="mc-region-442" style="position: absolute; top: 669pt; left: 354pt; width: 79.5pt; "><a name="" /><img border="0" src=".\addendum 4 optics telescopes_images/IMG0015_273747656.PNG" id="generatedImage442" path=".\addendum 4 optics telescopes_images/IMG0015_273747656.PNG" style="width: 79.5pt;height: 18.75pt;"></img></div><div id="mc-region-465" style="position: absolute; top: 704.25pt; left: 36pt; width: 74.75pt; "><a name="" /><p class="Style1">Calculate angle:</p></div><div id="mc-region-466" style="position: absolute; top: 705pt; left: 126pt; width: 42pt; "><a name="" /><img border="0" src=".\addendum 4 optics telescopes_images/IMG0016_273747656.PNG" id="generatedImage466" path=".\addendum 4 optics telescopes_images/IMG0016_273747656.PNG" style="width: 42pt;height: 12.75pt;"></img></div><div id="mc-region-467" style="position: absolute; top: 697.5pt; left: 198pt; width: 27.75pt; "><a name="" /><img border="0" src=".\addendum 4 optics telescopes_images/IMG0017_273747656.PNG" id="generatedImage467" path=".\addendum 4 optics telescopes_images/IMG0017_273747656.PNG" style="width: 27.75pt;height: 28.5pt;"></img></div><div id="mc-region-468" style="position: absolute; top: 704.25pt; left: 264pt; width: 60.75pt; background-color: inherit; "><a name="" /><img border="0" src=".\addendum 4 optics telescopes_images/IMG0018_273747687.PNG" id="generatedImage468" path=".\addendum 4 optics telescopes_images/IMG0018_273747687.PNG" style="width: 60.75pt;height: 13.5pt;"></img></div><div id="mc-region-474" style="position: absolute; top: 728.25pt; left: 30pt; width: 414.5pt; "><a name="" /><p class="Style1">4. A<b>ngular size, distance, small angle formula</b>. Astronomical objects are very far away, so their angular sizes are extermely small. The most commonly used unit to measure angular size is the arcsecond, defined in section 3 above. </p><p class="Style1"></p><p class="Style1">a. We define a<b> parsec</b> as:</p></div><div id="mc-region-475" style="position: absolute; top: 801pt; left: 96pt; width: 69.75pt; "><a name="" /><img border="0" src=".\addendum 4 optics telescopes_images/IMG0019_273747687.PNG" id="generatedImage475" path=".\addendum 4 optics telescopes_images/IMG0019_273747687.PNG" style="width: 69.75pt;height: 18.75pt;"></img></div><div id="mc-region-509" style="position: absolute; top: 836.25pt; left: 30pt; width: 170pt; "><a name="" /><p class="Style1">b. and an <b>astronomical unit (AU) </b>as:</p></div><div id="mc-region-478" style="position: absolute; top: 861pt; left: 102pt; width: 66.75pt; "><a name="" /><img border="0" src=".\addendum 4 optics telescopes_images/IMG0020_273747687.PNG" id="generatedImage478" path=".\addendum 4 optics telescopes_images/IMG0020_273747687.PNG" style="width: 66.75pt;height: 18.75pt;"></img></div><div id="mc-region-510" style="position: absolute; top: 890.25pt; left: 30pt; width: 408.5pt; "><a name="" /><p class="Style1">c. For small angles <span style="font-size: 10pt; font-family: 'Symbol', sans-serif;"><span style="font-weight: normal; font-style: normal;">a</span></span>, the tangent and the cos<span style="font-family: 'Arial', sans-serif;">ine</span> of an angle are (very nearly) equal to the angle itself (measured in radians) e.g.,</p></div><div id="mc-region-486" style="position: absolute; top: 926.25pt; left: 114pt; width: 45pt; "><a name="" /><img border="0" src=".\addendum 4 optics telescopes_images/IMG0021_273747687.PNG" id="generatedImage486" path=".\addendum 4 optics telescopes_images/IMG0021_273747687.PNG" style="width: 45pt;height: 13.5pt;"></img></div><div id="mc-region-487" style="position: absolute; top: 926.25pt; left: 174pt; width: 69.75pt; "><a name="" /><img border="0" src=".\addendum 4 optics telescopes_images/IMG0022_273747687.PNG" id="generatedImage487" path=".\addendum 4 optics telescopes_images/IMG0022_273747687.PNG" style="width: 69.75pt;height: 13.5pt;"></img></div><div id="mc-region-488" style="position: absolute; top: 926.25pt; left: 252pt; width: 69pt; "><a name="" /><img border="0" src=".\addendum 4 optics telescopes_images/IMG0023_273747703.PNG" id="generatedImage488" path=".\addendum 4 optics telescopes_images/IMG0023_273747703.PNG" style="width: 69pt;height: 13.5pt;"></img></div><div id="mc-region-490" style="position: absolute; top: 956.25pt; left: 48pt; width: 396.5pt; "><a name="" /><p class="Style1">d. The 'small angle formula' is a relation between the size of an object, its distance, and its angualr size. If the linear size <i>D</i> of an object is given in AU, and the angular size <span style="font-size: 10pt; font-family: 'Symbol', sans-serif;"><span style="font-weight: normal; font-style: normal;">a</span></span><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;"> </span></span>is in arcseconds, and the distance <i>d</i> is in parsecs, the relationship is simply written: </p></div><div id="mc-region-495" style="position: absolute; top: 1004.25pt; left: 204pt; width: 33.75pt; "><a name="" /><img border="0" src=".\addendum 4 optics telescopes_images/IMG0024_273747703.PNG" id="generatedImage495" path=".\addendum 4 optics telescopes_images/IMG0024_273747703.PNG" style="width: 33.75pt;height: 13.5pt;"></img></div><div id="mc-region-520" style="position: absolute; top: 1026pt; left: 90pt; width: 202.5pt; "><a name="" /><img border="0" src=".\addendum 4 optics telescopes_images/IMG0026_273747703.PNG" id="generatedImage520" path=".\addendum 4 optics telescopes_images/IMG0026_273747703.PNG" style="width: 202.5pt;height: 141.75pt;"></img></div><div id="mc-region-511" style="position: absolute; top: 1172.25pt; left: 24pt; width: 390.5pt; "><a name="" /><p class="Style1"><i>Example: An asteroid is observed to have an angular size </i><span style="font-size: 10pt; font-family: 'Symbol', sans-serif;"><i><span style="font-weight: normal;">a</span></i></span><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><i><span style="font-weight: normal;"> = 0.05 arcsec, and a distance of 3 AU. What is its angular size?</span></i></span></p></div><div id="mc-region-529" style="position: absolute; top: 1214.25pt; left: 54pt; width: 64.5pt; "><a name="" /><img border="0" src=".\addendum 4 optics telescopes_images/IMG0027_273747703.PNG" id="generatedImage529" path=".\addendum 4 optics telescopes_images/IMG0027_273747703.PNG" style="width: 64.5pt;height: 13.5pt;"></img></div><div id="mc-region-530" style="position: absolute; top: 1215pt; left: 144pt; width: 40.5pt; "><a name="" /><img border="0" src=".\addendum 4 optics telescopes_images/IMG0028_273747703.PNG" id="generatedImage530" path=".\addendum 4 optics telescopes_images/IMG0028_273747703.PNG" style="width: 40.5pt;height: 12.75pt;"></img></div><div id="mc-region-531" style="position: absolute; top: 1214.25pt; left: 204pt; width: 35.25pt; "><a name="" /><img border="0" src=".\addendum 4 optics telescopes_images/IMG0029_273747718.PNG" id="generatedImage531" path=".\addendum 4 optics telescopes_images/IMG0029_273747718.PNG" style="width: 35.25pt;height: 13.5pt;"></img></div><div id="mc-region-532" style="position: absolute; top: 1215pt; left: 264pt; width: 56.25pt; background-color: inherit; "><a name="" /><img border="0" src=".\addendum 4 optics telescopes_images/IMG0030_273747718.PNG" id="generatedImage532" path=".\addendum 4 optics telescopes_images/IMG0030_273747718.PNG" style="width: 56.25pt;height: 12.75pt;"></img></div><div></div></body></html>