<html><head><title>Math Addendum 2</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;; font-size: 10pt;font-weight: normal; font-style: normal; vertical-align: baseline; } p.Style2, li.Style2 /*Heading 1*/ { margin-left: 0pt; margin-right: 0pt; text-align: left; font-family: 'Arial', sans-serif;; font-size: 13.26315789473684pt;font-weight: bold; font-style: normal; vertical-align: baseline; } p.Style3, li.Style3 /*Heading 2*/ { margin-left: 0pt; margin-right: 0pt; text-align: left; font-family: 'Arial', sans-serif;; font-size: 12pt;font-weight: bold; font-style: italic; vertical-align: baseline; } p.Style4, li.Style4 /*Heading 3*/ { margin-left: 0pt; margin-right: 0pt; text-align: left; font-family: 'Arial', sans-serif;; font-size: 12pt;font-weight: normal; font-style: normal; vertical-align: baseline; } p.Style5, li.Style5 /*Paragraph*/ { margin-left: 0pt; margin-right: 0pt; text-indent: 21pt; text-align: left; font-family: 'Arial', sans-serif;; font-size: 10pt;font-weight: normal; font-style: normal; vertical-align: baseline; } p.Style6, li.Style6 /*List*/ { margin-left: 14.25pt; margin-right: 0pt; text-indent: -14.25pt; text-align: left; font-family: 'Arial', sans-serif;; font-size: 10pt;font-weight: normal; font-style: normal; vertical-align: baseline; } p.Style7, li.Style7 /*Indent*/ { margin-left: 108pt; margin-right: 0pt; text-align: left; font-family: 'Arial', sans-serif;; font-size: 10pt;font-weight: normal; font-style: normal; vertical-align: baseline; } p.Style8, li.Style8 /*Title*/ { margin-left: 0pt; margin-right: 0pt; text-align: center; font-family: 'Times New Roman', sans-serif;; font-size: 24pt;font-weight: bold; font-style: normal; vertical-align: baseline; } p.Style9, li.Style9 /*Subtitle*/ { margin-left: 0pt; margin-right: 0pt; text-align: center; font-family: 'Times New Roman', sans-serif;; font-size: 18pt;font-weight: normal; font-style: normal; vertical-align: baseline; } </style></head><body><div id="mc-region-2" style="position: absolute; top: 24pt; left: 18pt; width: 273.75pt; "><a name="" /><img border="0" src=".\addendum 2 synodic-sidereal periods cel coords_images/IMG0003_105338296.PNG" id="generatedImage2" path=".\addendum 2 synodic-sidereal periods cel coords_images/IMG0003_105338296.PNG" style="width: 273.75pt;height: 41.25pt;"></img></div><div id="mc-region-5" style="position: absolute; top: 86.25pt; left: 18pt; width: 111.5pt; "><a name="" /><p class="Style1"><b>MathCAD Addendum 2</b></p><p class="Style1"><i>Version: 12 July 2006</i></p></div><div id="mc-region-39" style="position: absolute; top: 128.25pt; left: 12pt; width: 223.25pt; "><a name="" /><p class="Style1">1. <b><i>Synodic and Sidereal Periods</i></b></p></div><div id="mc-region-97" style="position: absolute; top: 152.25pt; left: 12pt; width: 419pt; "><a name="" /><p class="Style1">The interval between sucessive identical phases of a planet or of the Moon, as viewed from Earth, is the called the <b>synodic</b> period. For example, the interval (period) between successive full Moon phases is 29.53 days. This is <b><i>not</i></b> the same as the true orbital or <b>sidereal</b> period, which is 27.3 days. The term sidereal means 'with respect to the stars'. i.e. indepenent of the orbital motion of the Earth, which causes the difference between sidereal and synodic period, as explined in the lecture. The relationship is either:</p></div><div id="mc-region-128" style="position: absolute; top: 242.25pt; left: 24pt; width: 132pt; "><a name="" /><img border="0" src=".\addendum 2 synodic-sidereal periods cel coords_images/IMG0004_105338312.PNG" id="generatedImage128" path=".\addendum 2 synodic-sidereal periods cel coords_images/IMG0004_105338312.PNG" style="width: 132pt;height: 31.5pt;"></img></div><div id="mc-region-126" style="position: absolute; top: 254.25pt; left: 180pt; width: 221.75pt; "><a name="" /><p class="Style1">for inferior planets (Mercury, Venus) and the Moon</p></div><div id="mc-region-124" style="position: absolute; top: 290.25pt; left: 36pt; width: 16.25pt; "><a name="" /><p class="Style1">or,</p></div><div id="mc-region-127" style="position: absolute; top: 308.25pt; left: 24pt; width: 132.75pt; "><a name="" /><img border="0" src=".\addendum 2 synodic-sidereal periods cel coords_images/IMG0005_105338328.PNG" id="generatedImage127" path=".\addendum 2 synodic-sidereal periods cel coords_images/IMG0005_105338328.PNG" style="width: 132.75pt;height: 31.5pt;"></img></div><div id="mc-region-125" style="position: absolute; top: 314.25pt; left: 180pt; width: 175.25pt; "><a name="" /><p class="Style1">for superior planets (Mars, Jupiter, etc) </p></div><div id="mc-region-129" style="position: absolute; top: 356.25pt; left: 18pt; width: 269pt; "><a name="" /><p class="Style1">Where P<sub>Earth</sub> = 365.25 days is the orbital period of the Earth.</p></div><div id="mc-region-114" style="position: absolute; top: 386.25pt; left: 12pt; width: 287pt; "><a name="" /><p class="Style1">For example, let's use the Moon and solve for the sidereal period.</p></div><div id="mc-region-102" style="position: absolute; top: 410.25pt; left: 120pt; width: 167.25pt; "><a name="" /><img border="0" src=".\addendum 2 synodic-sidereal periods cel coords_images/IMG0006_105338343.PNG" id="generatedImage102" path=".\addendum 2 synodic-sidereal periods cel coords_images/IMG0006_105338343.PNG" style="width: 167.25pt;height: 33.75pt;"></img></div><div id="mc-region-105" style="position: absolute; top: 459pt; left: 126pt; width: 84pt; background-color: inherit; "><a name="" /><img border="0" src=".\addendum 2 synodic-sidereal periods cel coords_images/IMG0007_105338359.PNG" id="generatedImage105" path=".\addendum 2 synodic-sidereal periods cel coords_images/IMG0007_105338359.PNG" style="width: 84pt;height: 16.5pt;"></img></div><div id="mc-region-109" style="position: absolute; top: 482.25pt; left: 6pt; width: 421.25pt; "><a name="" /><p class="Style1">A second example is Mars. If the observed interval between successive oppositions is 780 days, what is the true (sidereal) orbital period of Mars? </p></div><div id="mc-region-111" style="position: absolute; top: 524.25pt; left: 120pt; width: 159.75pt; "><a name="" /><img border="0" src=".\addendum 2 synodic-sidereal periods cel coords_images/IMG0008_105338375.PNG" id="generatedImage111" path=".\addendum 2 synodic-sidereal periods cel coords_images/IMG0008_105338375.PNG" style="width: 159.75pt;height: 33.75pt;"></img></div><div id="mc-region-113" style="position: absolute; top: 585pt; left: 132pt; width: 88.5pt; background-color: inherit; "><a name="" /><img border="0" src=".\addendum 2 synodic-sidereal periods cel coords_images/IMG0009_105338390.PNG" id="generatedImage113" path=".\addendum 2 synodic-sidereal periods cel coords_images/IMG0009_105338390.PNG" style="width: 88.5pt;height: 16.5pt;"></img></div><div id="mc-region-130" style="position: absolute; top: 626.25pt; left: 6pt; width: 247.25pt; "><a name="" /><p class="Style1"><b>2. </b><b>Celestial coordinate system, angular separations</b></p></div><div id="mc-region-132" style="position: absolute; top: 656.25pt; left: 12pt; width: 402.5pt; "><a name="" /><p class="Style1" style="margin-left: 0pt; margin-right: 0pt; text-align: left;">The position of celestial objects (planets, stars, galaxies, etc) on the celestial sphere is given using a grid system very similar to the latitude-longitude system used on for places on the Earth's surface. The analogue to latitude is <b>declination</b> (written <span style="font-size: 10pt; font-family: 'Symbol', sans-serif;"><span style="font-weight: normal; font-style: normal;">d</span></span><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;">) </span></span>and the analogue to longitude is <b>right ascension </b><span style="font-weight: normal;">(written </span><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>. Like latitude, declination is measured from -90 deg to +90 deg, wtih +90 deg being at the north celestial pole, very close to the north star Polaris. Declination subdivisions are in arcminutes and arcseconds (see MathCAD worksheet 1). However, right ascension is measured in time-like angular units: The 360 degree circle is divided into 24 hours (15 deg = 1 hr), with sub-divisions into minutes and seconds. Hence a right ascension of 90 deg = 6:00:00 hrs while a right ascension of 187.5 deg = 12:30:00. There is a MathCAD dimension unit that can convert from deg to hhmmss format: </p></div><div id="mc-region-134" style="position: absolute; top: 800.25pt; left: 108pt; width: 159.75pt; "><a name="" /><img border="0" src=".\addendum 2 synodic-sidereal periods cel coords_images/IMG0010_105338406.PNG" id="generatedImage134" path=".\addendum 2 synodic-sidereal periods cel coords_images/IMG0010_105338406.PNG" style="width: 159.75pt;height: 27.75pt;"></img></div><div id="mc-region-135" style="position: absolute; top: 842.25pt; left: 18pt; width: 379.25pt; "><a name="" /><p class="Style1">Example 1: Two stars have celestial coordinates <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>08:14:13, <span style="font-size: 10pt; font-family: 'Symbol', sans-serif;"><span style="font-weight: normal; font-style: normal;">d</span></span><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;"> = </span></span>+02:12:41 and <span style="font-size: 10pt; font-family: 'Symbol', sans-serif;"><span style="font-weight: normal; font-style: normal;">a</span></span><span style="font-weight: normal; font-style: normal;"><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"> = </span></span>09:01:12, <span style="font-size: 10pt; font-family: 'Symbol', sans-serif;"><span style="font-weight: normal; font-style: normal;">d</span></span><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;"> = </span></span> -11:13:24. What is the angular distance (in degrees) between the stars?</p><p class="Style1"></p><p class="Style1">Ans: Use the MathCAD function DMS to convert to radians, using a 3x1 matrix with each argument to DMS (multiplying by 15 for the right ascension since 1hr = 15 deg):</p></div><div id="mc-region-154" style="position: absolute; top: 928.5pt; left: 30pt; width: 87.75pt; "><a name="" /><img border="0" src=".\addendum 2 synodic-sidereal periods cel coords_images/IMG0011_105338437.PNG" id="generatedImage154" path=".\addendum 2 synodic-sidereal periods cel coords_images/IMG0011_105338437.PNG" style="width: 87.75pt;height: 45.75pt;"></img></div><div id="mc-region-156" style="position: absolute; top: 944.25pt; left: 138pt; width: 48.75pt; "><a name="" /><img border="0" src=".\addendum 2 synodic-sidereal periods cel coords_images/IMG0012_105338437.PNG" id="generatedImage156" path=".\addendum 2 synodic-sidereal periods cel coords_images/IMG0012_105338437.PNG" style="width: 48.75pt;height: 13.5pt;"></img></div><div id="mc-region-139" style="position: absolute; top: 928.5pt; left: 234pt; width: 78pt; "><a name="" /><img border="0" src=".\addendum 2 synodic-sidereal periods cel coords_images/IMG0013_105338453.PNG" id="generatedImage139" path=".\addendum 2 synodic-sidereal periods cel coords_images/IMG0013_105338453.PNG" style="width: 78pt;height: 45.75pt;"></img></div><div id="mc-region-159" style="position: absolute; top: 944.25pt; left: 342pt; width: 46.5pt; "><a name="" /><img border="0" src=".\addendum 2 synodic-sidereal periods cel coords_images/IMG0014_105338484.PNG" id="generatedImage159" path=".\addendum 2 synodic-sidereal periods cel coords_images/IMG0014_105338484.PNG" style="width: 46.5pt;height: 13.5pt;"></img></div><div id="mc-region-155" style="position: absolute; top: 982.5pt; left: 30pt; width: 87.75pt; "><a name="" /><img border="0" src=".\addendum 2 synodic-sidereal periods cel coords_images/IMG0015_105338500.PNG" id="generatedImage155" path=".\addendum 2 synodic-sidereal periods cel coords_images/IMG0015_105338500.PNG" style="width: 87.75pt;height: 45.75pt;"></img></div><div id="mc-region-158" style="position: absolute; top: 998.25pt; left: 144pt; width: 48.75pt; "><a name="" /><img border="0" src=".\addendum 2 synodic-sidereal periods cel coords_images/IMG0016_105338500.PNG" id="generatedImage158" path=".\addendum 2 synodic-sidereal periods cel coords_images/IMG0016_105338500.PNG" style="width: 48.75pt;height: 13.5pt;"></img></div><div id="mc-region-146" style="position: absolute; top: 982.5pt; left: 234pt; width: 83.25pt; "><a name="" /><img border="0" src=".\addendum 2 synodic-sidereal periods cel coords_images/IMG0017_105338515.PNG" id="generatedImage146" path=".\addendum 2 synodic-sidereal periods cel coords_images/IMG0017_105338515.PNG" style="width: 83.25pt;height: 45.75pt;"></img></div><div id="mc-region-160" style="position: absolute; top: 998.25pt; left: 336pt; width: 51.75pt; "><a name="" /><img border="0" src=".\addendum 2 synodic-sidereal periods cel coords_images/IMG0018_105338531.PNG" id="generatedImage160" path=".\addendum 2 synodic-sidereal periods cel coords_images/IMG0018_105338531.PNG" style="width: 51.75pt;height: 13.5pt;"></img></div><div id="mc-region-153" style="position: absolute; top: 1040.25pt; left: 18pt; width: 401pt; "><a name="" /><p class="Style1">Since the right ascension and declination axes are orthogonal, use Pythagorean theorem to determine the separation:</p></div><div id="mc-region-148" style="position: absolute; top: 1076.25pt; left: 132pt; width: 58.5pt; "><a name="" /><img border="0" src=".\addendum 2 synodic-sidereal periods cel coords_images/IMG0019_105338546.PNG" id="generatedImage148" path=".\addendum 2 synodic-sidereal periods cel coords_images/IMG0019_105338546.PNG" style="width: 58.5pt;height: 13.5pt;"></img></div><div id="mc-region-149" style="position: absolute; top: 1076.25pt; left: 222pt; width: 51.75pt; "><a name="" /><img border="0" src=".\addendum 2 synodic-sidereal periods cel coords_images/IMG0020_105338562.PNG" id="generatedImage149" path=".\addendum 2 synodic-sidereal periods cel coords_images/IMG0020_105338562.PNG" style="width: 51.75pt;height: 13.5pt;"></img></div><div id="mc-region-150" style="position: absolute; top: 1098.75pt; left: 144pt; width: 76.5pt; "><a name="" /><img border="0" src=".\addendum 2 synodic-sidereal periods cel coords_images/IMG0021_105338578.PNG" id="generatedImage150" path=".\addendum 2 synodic-sidereal periods cel coords_images/IMG0021_105338578.PNG" style="width: 76.5pt;height: 21pt;"></img></div><div id="mc-region-151" style="position: absolute; top: 1106.25pt; left: 252pt; width: 65.25pt; background-color: inherit; "><a name="" /><img border="0" src=".\addendum 2 synodic-sidereal periods cel coords_images/IMG0022_105338593.PNG" id="generatedImage151" path=".\addendum 2 synodic-sidereal periods cel coords_images/IMG0022_105338593.PNG" style="width: 65.25pt;height: 13.5pt;"></img></div><div id="mc-region-162" style="position: absolute; top: 1136.25pt; left: 24pt; width: 11pt; "><a name="" /><p class="Style1"> </p></div><div id="mc-region-164" style="position: absolute; top: 1142.25pt; left: 24pt; width: 126.5pt; "><a name="" /><p class="Style1"><b>3. Coordinate Precession.</b></p></div><div></div></body></html>