7268 /www/nrich/html/content/id/7268/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><ul id="stemLinks"> <li><a href="http://nrich.maths.org/7278">Warm-up</a></li> <li><a href="http://nrich.maths.org/7428">Try this next</a></li> <li><a href="http://nrich.maths.org/6683">Think higher</a></li> <li><a href="http://plus.maths.org/content/how-does-gravity-work">Read: mathematics</a></li> <li><a href="http://scaleofuniverse.com/">Read: science</a></li> <li><a href="http://motivate.maths.org/conferences/conference.php?conf_id=86">Explore further</a></li> </ul> <div> </div> <br></br> <br></br> <p>The following table comprises real astronomical data (compiled from Wikipedia) which describe the elliptical paths taken by some key objects in our solar system:<br></br>  </p> <table style="" border="1"> <tbody> <tr> <td align="center" style="">Name</td> <td align="center" style="">Diameter relative to Earth</td> <td align="center" style="">Average distance from the sun (in AUs)</td> <td align="center" style="">Time to orbit the sun (years)</td> <td align="center" style="">Inclination of orbit to sun&#39;s equator (in degrees)</td> <td align="center" style=""> <div>Orbital Eccentricity</div> <div>e</div> </td> <td align="center" style="">Time to spin on axis (days)</td> </tr> <tr> <td align="center" style="">The Sun</td> <td align="center" style="">109</td> <td align="center" style="">--</td> <td align="center" style="">--</td> <td align="center" style="">--</td> <td align="center" style="">--</td> <td align="center" style="">26.38</td> </tr> <tr> <td align="center" style="">Mercury</td> <td align="center" style="">0.382</td> <td align="center" style="">0.39</td> <td align="center" style="">0.24</td> <td align="center" style="">3.38</td> <td align="center" style="">0.206</td> <td align="center" style="">58.64</td> </tr> <tr> <td align="center" style="">Venus</td> <td align="center" style="">0.949</td> <td align="center" style="">0.72</td> <td align="center" style="">0.62</td> <td align="center" style="">3.86</td> <td align="center" style="">0.007</td> <td align="center" style="">-243.02</td> </tr> <tr> <td align="center" style="">Earth</td> <td align="center" style="">1.00</td> <td align="center" style="">1.00</td> <td align="center" style="">1.00</td> <td align="center" style="">7.25</td> <td align="center" style="">0.017</td> <td align="center" style="">1.00</td> </tr> <tr> <td align="center" style="">Mars</td> <td align="center" style="">0.532</td> <td align="center" style="">1.52</td> <td align="center" style="">1.88</td> <td align="center" style="">5.65</td> <td align="center" style="">0.093</td> <td align="center" style="">1.03</td> </tr> <tr> <td align="center" style="">Jupiter</td> <td align="center" style="">11.209</td> <td align="center" style="">5.20</td> <td align="center" style="">11.86</td> <td align="center" style="">6.09</td> <td align="center" style="">0.048</td> <td align="center" style="">0.41</td> </tr> <tr> <td align="center" style="">Saturn</td> <td align="center" style="">9.449</td> <td align="center" style="">9.54</td> <td align="center" style="">29.46</td> <td align="center" style="">5.51</td> <td align="center" style="">0.054</td> <td align="center" style="">0.43</td> </tr> <tr> <td align="center" style="">Uranus</td> <td align="center" style="">4.007</td> <td align="center" style="">19.22</td> <td align="center" style="">84.01</td> <td align="center" style="">6.48</td> <td align="center" style="">0.047</td> <td align="center" style="">-0.72</td> </tr> <tr> <td align="center" style="">Neptune</td> <td align="center" style="">3.883</td> <td align="center" style="">30.06</td> <td align="center" style="">164.8</td> <td align="center" style="">6.43</td> <td align="center" style="">0.009</td> <td align="center" style="">0.67</td> </tr> </tbody> </table> <br></br> <p>The actual numbers for the earth are</p> <table style="" border="1"> <tbody> <tr> <td>Diameter</td> <td>Mass kg</td> <td>Distance from sun</td> <td>Orbital period</td> <td>Rotation time</td> </tr> <tr> <td>12756 km</td> <td>5.9736 E10</td> <td>147.1-152.1 million km</td> <td>365.256366 days</td> <td>23 hours 56 minutes</td> </tr> </tbody> </table> <br></br> <br></br> <p>Make an accurate top down drawing of the solar system on, for example, a piece of A4 paper.<br></br>  <br></br> You will first need to try to make sense of the data in the table!<br></br> <br></br> You could either assume that the orbits are all circular, centred on the sun. Or (a lot more tricky) you could take into account the eccentricity of the orbit: $e = \frac{r_{max}-r_{min}}{r_{max}+r_{min}}$ where $r_{max}$ and $r_{min}$ are the maximum and minimum distances from the sun respectively.<br></br>  <br></br> Once you have made your drawing, you can make a side-view to show the effects of inclination of the orbits.<br></br> <br></br>  <br></br> In late April 2002 a <span style="font-style: italic;">grand conjunction</span> occurred in which Mercury, Venus, Mars, Saturn and Jupiter were all visible from various places on Earth at night. What does this tell us about the possible range of locations of these planets of their orbits?<br></br>  <br></br> From this information, for which of the planets could you meaningfully predict their locations relative to earth now?<br></br>  <br></br> <span style="font-style: italic;">Extension:Estimate how often you would expect Mercury, Venus, Mars, Saturn and Jupiter to line up. How often might you expect all of the planets to line up</span>?<br></br>  <br></br>  </p> <div class="framework">NOTES AND BACKGROUND<br></br> <br></br> Planetary conjunctions are beautiful to observe, as planets are the brightest &#39;stars&#39; in the sky. You can see an image of a planetery conjunction <a href="http://apod.nasa.gov/apod/ap080310.html">here</a>. <br></br>  </div> <p> </p></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">There are many diagrams of the solar system available on the internet.<br></br> <br></br> <a href="http://ssd.jpl.nasa.gov/?orbits">This page</a> from NASA&#39;s website might be a good place to start.</mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">It is not envisaged that <a href="http://nrich.maths.org/7268">this problem</a> would be used as a class problem.  It is more appropriate for an enthusiastic student or small group of students looking for a challenge to work on independently.</mdoxml> 2 3 0 0 0 1 1 Make an accurate diagram of the solar system and explore the concept of a grand conjunction. 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