82 
POLYTROPIC GAS SPHERES 
Equations (55-41) and (55-42) show that for different points in the same 
star p oc cf) n and P oc pcf> oc <f) n+1 . Hence 
<f>M>o = u, p/p 0 — u n , P/P 0 = u n + 1 (56-1). 
Thus columns 2, 3, 4 of the tables give the potential, density and pressure 
in terms of the central potential, density and pressure*. 
Poly tropic Solutions. 
Table 4. 
(n = 2, y = 1-5.) 
z 
u 
u n 
- du/dz 
-zdz/Sdu 
-z 2 du/dz 
0-00 
1-00000 
1-00000 
1-00000 
•00000 
1-0000 
•0000 
0-25 
•98969 
•97950 
•96940 
•08247 
1-0105 
•0052 
0-50 
•95937 
•92040 
•88302 
•15865 
1-0750 
•0388 
0-75 
•91128 
•83042 
•75675 
•22386 
1-1168 
•1259 
1-00 
•84864 
•72018 
•61117 
•27453 
1-2142 
•2745 
1-25 
•77533 
•60114 
•46609 
•30937 
1-3468 
•4834 
1-50 
•69531 
•48346 
•33615 
•32825 
1-5232 
•7386 
1-75 
•61238 
•37501 
•22965 
•33307 
1-7514 
1-0200 
2-00 
•52974 
•28062 
•14866 
•32640 
2-0425 
1-3056 
2-50 
•37463 
•14035 
•05258 
•29023 
2-8713 
1-8140 
3-00 
•24166 
•05840 
•01411 
•24067 
4-1550 
2-1660 
3-50 
•13379 
•01790 
•002395 
•19169 
6-0863 
2-3482 
4-00 
•04866 
•002368 
•000115 
•15040 
8-8653 
2-4064 
4-25 
•01326 
•000176 
•000002 
•13346 
10-615 
2-4106 
4-3518 
•00000 
•000000 
•000000 
•12729 
11-396 
2-4107 
Table 5. 
(n = 2-5, y = 1-4.) 
z 
u 
u n 
u "+ 1 
— dujdz 
— zdz/3du 
- z 2 dufdz 
0-00 
1-00000 
1-00000 
1-00000 
•00000 
1-0000 
•0000 
0-25 
•98971 
•97450 
•96447 
•08226 
10130 
•0051 
0-50 
•95961 
•90202 
•86560 
•15676 
1-0632 
•0392 
0-75 
•91242 
•79520 
•72555 
•21798 
11469 
•1226 
1-00 
•85196 
•66997 
•57079 
•26282 
1-2683 
•2628 
1-25 
•78246 
•54156 
•42375 
•29036 
1-4350 
•4537 
1-50 
•70809 
•42192 
•29876 
•30213 
1-6549 
•6798 
1-75 
•63246 
•31811 
•20119 
•29532 
1-9753 
•9044 
2-00 
•55961 
•23428 
•13111 
•28614 
2-3298 
11446 
2-50 
•42473 
•11756 
•04993 
•25080 
3-3227 
1-5675 
3-00 
•31000 
•05351 
•01659 
•20793 
4-8093 
1-8714 
3-50 
•21752 
•02207 
•004800 
•16783 
6-9517 
2-0560 
4-00 
•14300 
■007733 
•001106 
•13445 
9-9170 
2-1512 
4-50 
•08263 
•001963 
■000162 
•10813 
10-539 
2-1896 
5-00 
•03384 
•000211 
•000007 
•08796 
18-948 
2-1990 
5-4172 
•00000 
•000000 
•000000 
•07500 
24-076 
2-2010 
* In this Chapter we denote central values by the suffix 0; elsewhere in the book 
the suffix c is generally used.