A polytropic profile is a function of the form

where is the so-called polytropic index. Polytropic profiles obey the Lane-Emden equation and have profiles of the form

together with the boundary conditions


<P> <DIV ALIGN="CENTER"> <TABLE CELLPADDING=3 BORDER="1"> <TR><TD ALIGN="LEFT"><i>n</i></TD> <TD ALIGN="RIGHT"><IMG WIDTH="18" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="http://scienceworld.wolfram.com/physics/pimg710.gif"></TD> <TD ALIGN="LEFT"><IMG WIDTH="20" HEIGHT="28" ALIGN="MIDDLE" BORDER="0" SRC="http://scienceworld.wolfram.com/physics/pimg711.gif"></TD> <TD ALIGN="LEFT"><IMG WIDTH="21" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="http://scienceworld.wolfram.com/physics/pimg712.gif"></TD> <TD ALIGN="LEFT"><IMG WIDTH="13" HEIGHT="28" ALIGN="MIDDLE" BORDER="0" SRC="http://scienceworld.wolfram.com/physics/pimg45.gif"></TD> <TD ALIGN="LEFT"><IMG WIDTH="21" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="http://scienceworld.wolfram.com/physics/pimg429.gif"></TD> <TD ALIGN="LEFT"><i>U</i></TD> </TR> <TR><TD ALIGN="LEFT">&nbsp;</TD> <TD ALIGN="RIGHT">&nbsp;</TD> <TD ALIGN="LEFT"><IMG WIDTH="24" HEIGHT="35" ALIGN="MIDDLE" BORDER="0" SRC="http://scienceworld.wolfram.com/physics/pimg713.gif"></TD> <TD ALIGN="LEFT"> <IMG WIDTH="37" HEIGHT="40" ALIGN="MIDDLE" BORDER="0" SRC="http://scienceworld.wolfram.com/physics/pimg714.gif"></TD> <TD ALIGN="LEFT"><IMG WIDTH="37" HEIGHT="34" ALIGN="MIDDLE" BORDER="0" SRC="http://scienceworld.wolfram.com/physics/pimg715.gif"></TD> <TD ALIGN="LEFT"> <IMG WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0" SRC="http://scienceworld.wolfram.com/physics/pimg716.gif"></TD> <TD ALIGN="LEFT"> <IMG WIDTH="50" HEIGHT="40" ALIGN="MIDDLE" BORDER="0" SRC="http://scienceworld.wolfram.com/physics/pimg717.gif"></TD> </TR> <TR><TD ALIGN="LEFT">0</TD> <TD ALIGN="RIGHT"><IMG WIDTH="25" HEIGHT="36" ALIGN="MIDDLE" BORDER="0" SRC="http://scienceworld.wolfram.com/physics/pimg718.gif"></TD> <TD ALIGN="LEFT"> <IMG WIDTH="22" HEIGHT="34" ALIGN="MIDDLE" BORDER="0" SRC="http://scienceworld.wolfram.com/physics/pimg719.gif"></TD> <TD ALIGN="LEFT"> <IMG WIDTH="28" HEIGHT="34" ALIGN="MIDDLE" BORDER="0" SRC="http://scienceworld.wolfram.com/physics/pimg720.gif"></TD> <TD ALIGN="LEFT"> <IMG WIDTH="14" HEIGHT="34" ALIGN="MIDDLE" BORDER="0" SRC="http://scienceworld.wolfram.com/physics/pimg721.gif"></TD> <TD ALIGN="LEFT"> <IMG WIDTH="14" HEIGHT="34" ALIGN="MIDDLE" BORDER="0" SRC="http://scienceworld.wolfram.com/physics/pimg722.gif"></TD> <TD ALIGN="LEFT"> <IMG WIDTH="21" HEIGHT="34" ALIGN="MIDDLE" BORDER="0" SRC="http://scienceworld.wolfram.com/physics/pimg723.gif"></TD> </TR> <TR><TD ALIGN="LEFT">1</TD> <TD ALIGN="RIGHT"><IMG WIDTH="14" HEIGHT="14" ALIGN="BOTTOM" BORDER="0" SRC="http://scienceworld.wolfram.com/physics/pimg724.gif"></TD> <TD ALIGN="LEFT"><IMG WIDTH="16" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="http://scienceworld.wolfram.com/physics/pimg725.gif"></TD> <TD ALIGN="LEFT"><IMG WIDTH="16" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="http://scienceworld.wolfram.com/physics/pimg726.gif"></TD> <TD ALIGN="LEFT"> <IMG WIDTH="55" HEIGHT="43" ALIGN="MIDDLE" BORDER="0" SRC="http://scienceworld.wolfram.com/physics/pimg727.gif"></TD> <TD ALIGN="LEFT"> <IMG WIDTH="14" HEIGHT="34" ALIGN="MIDDLE" BORDER="0" SRC="http://scienceworld.wolfram.com/physics/pimg722.gif"></TD> <TD ALIGN="LEFT"> <IMG WIDTH="14" HEIGHT="34" ALIGN="MIDDLE" BORDER="0" SRC="http://scienceworld.wolfram.com/physics/pimg728.gif"></TD> </TR> <TR><TD ALIGN="LEFT"> <IMG WIDTH="14" HEIGHT="34" ALIGN="MIDDLE" BORDER="0" SRC="http://scienceworld.wolfram.com/physics/pimg729.gif"></TD> <TD ALIGN="RIGHT">3.653</TD> <TD ALIGN="LEFT">&nbsp;</TD> <TD ALIGN="LEFT">&nbsp;</TD> <TD ALIGN="LEFT">&nbsp;</TD> <TD ALIGN="LEFT">&nbsp;</TD> <TD ALIGN="LEFT">&nbsp;</TD> </TR> <TR><TD ALIGN="LEFT">2</TD> <TD ALIGN="RIGHT">4.353</TD> <TD ALIGN="LEFT">1.12</TD> <TD ALIGN="LEFT">0.55</TD> <TD ALIGN="LEFT">0.24</TD> <TD ALIGN="LEFT">0.49</TD> <TD ALIGN="LEFT">0.40</TD> </TR> <TR><TD ALIGN="LEFT"> <IMG WIDTH="14" HEIGHT="34" ALIGN="MIDDLE" BORDER="0" SRC="http://scienceworld.wolfram.com/physics/pimg730.gif"></TD> <TD ALIGN="RIGHT">5.355</TD> <TD ALIGN="LEFT">&nbsp;</TD> <TD ALIGN="LEFT">&nbsp;</TD> <TD ALIGN="LEFT">&nbsp;</TD> <TD ALIGN="LEFT">&nbsp;</TD> <TD ALIGN="LEFT">&nbsp;</TD> </TR> <TR><TD ALIGN="LEFT">3</TD> <TD ALIGN="RIGHT">6.896</TD> <TD ALIGN="LEFT">1.84</TD> <TD ALIGN="LEFT">0.83</TD> <TD ALIGN="LEFT">0.22</TD> <TD ALIGN="LEFT">0.45</TD> <TD ALIGN="LEFT">0.42</TD> </TR> <TR><TD ALIGN="LEFT"> <IMG WIDTH="14" HEIGHT="34" ALIGN="MIDDLE" BORDER="0" SRC="http://scienceworld.wolfram.com/physics/pimg731.gif"></TD> <TD ALIGN="RIGHT">9.535</TD> <TD ALIGN="LEFT">&nbsp;</TD> <TD ALIGN="LEFT">&nbsp;</TD> <TD ALIGN="LEFT">&nbsp;</TD> <TD ALIGN="LEFT">&nbsp;</TD> <TD ALIGN="LEFT">&nbsp;</TD> </TR> <TR><TD ALIGN="LEFT">4</TD> <TD ALIGN="RIGHT">14.97</TD> <TD ALIGN="LEFT">4.16</TD> <TD ALIGN="LEFT">1.48</TD> <TD ALIGN="LEFT">0.21</TD> <TD ALIGN="LEFT">0.36</TD> <TD ALIGN="LEFT">0.44</TD> </TR> <TR><TD ALIGN="LEFT"> <IMG WIDTH="14" HEIGHT="34" ALIGN="MIDDLE" BORDER="0" SRC="http://scienceworld.wolfram.com/physics/pimg732.gif"></TD> <TD ALIGN="RIGHT">31.83</TD> <TD ALIGN="LEFT">&nbsp;</TD> <TD ALIGN="LEFT">&nbsp;</TD> <TD ALIGN="LEFT">&nbsp;</TD> <TD ALIGN="LEFT">&nbsp;</TD> <TD ALIGN="LEFT">&nbsp;</TD> </TR> <TR><TD ALIGN="LEFT">5</TD> <TD ALIGN="RIGHT"><IMG WIDTH="20" HEIGHT="14" ALIGN="BOTTOM" BORDER="0" SRC="http://scienceworld.wolfram.com/physics/pimg184.gif"></TD> <TD ALIGN="LEFT">&nbsp;</TD> <TD ALIGN="LEFT">&nbsp;</TD> <TD ALIGN="LEFT">&nbsp;</TD> <TD ALIGN="LEFT">&nbsp;</TD> <TD ALIGN="LEFT">&nbsp;</TD> </TR> </TABLE> </DIV>
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