90
POLYTROPIC GAS SPHERES
Then (63-3) reduces to the standard form
d 2 u 2 du
dz 2 ^ z dz 6
= 0
(63-5),
with the central conditions u = 0, du/dz = 0.
The solution calculated by Emden is given in Table 7. The successive
columns are proportional to (1) distance from centre, (2) potential,
(3) density and pressure, (4) acceleration of gravity, (5) central density
divided by mean density interior to r, (6) mass interior to r.
Table 7.
Isothermal Gas Sphere.
z
- U
gU
- du/dz
-zdz/3du
- z 2 du/dz
0-00
•00000
1-00000
•00000
1-000
•0000
0-25
•01037
•98969
•08290
1-005
•0052
0-50
•04113
•95971
•16225
1-027
•0406
0-75
•09113
•91290
•23819
1-050
•1340
1-00
•15903
•85296
•30370
1-097
•3037
1-25
•24225
•78486
•36045
1-156
•5632
1-50
•33847
•71285
•40432
1-237
•9097
1-75
•44488
•64090
•44390
1-314
1-3595
2-00
•55967
•57140
•47286
1-410
1-8914
2-5
•80584
•44671
•50694
1-644
3-1684
3
1-06226
•34537
•51625
1-937
4-6462
3-5
1-31937
•26730
•51006
2-287
6-2483
4
1-57071
•20790
•49403
2-699
7-9045
4-5
1-81246
•16325
•47234
3-176
9-5650
5
2-04264
•12968
•44813
3-719
11-203
6
2-46598
•08493
•39879
5-015
14-353
7
2-84160
•05833
•35334
6-604
17-214
8
3-17489
•04180
•31372
8-500
20-078
9
3-47128
•03108
•27989
10-718
22-670
10
3-73646
•02384
•25121
13-269
25-121
100
8-59506
•000175
•01843
1808-6
184-3
1000
13-09847
•000002
•002045
163000
2045-1
Minimal Problems.
64. Up to the present we have restricted the investigation to distribu
tions arranged according to a poly tropic model. In the following theorems
no such restriction is imposed, but we postulate that the density does not
decrease as we go inwards. The new limitation is scarcely likely to exclude
any case deserving serious consideration in application to the stars. It
is not likely that a distribution could be stable with dp I dr positive.
In Problem II we shall also postulate that the temperature does not
decrease inwards. This limitation is entirely innocuous. A positive value
