darwin and modern science-第169节

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Stars　and　planets　are　formed　of　materials　which　yield　to　the　enormous　forces　called　into　play　by　gravity　and　rotation。　　This　is　obviously　true　if　they　are　gaseous　or　fluid；　and　even　solid　matter　becomes　plastic　under　sufficiently　great　stresses。　　Nothing　approaching　a　complete　study　of　the　equilibrium　of　a　heterogeneous　star　has　yet　been　found　possible；　and　we　are　driven　to　consider　only　bodies　of　simpler　construction。　　I　shall　begin　therefore　by　explaining　what　is　known　about　the　shapes　which　may　be　assumed　by　a　mass　of　incompressible　liquid　of　uniform　density　under　the　influences　of　gravity　and　of　rotation。　　Such　a　liquid　mass　may　be　regarded　as　an　ideal　star；　which　resembles　a　real　star　in　the　fact　that　it　is　formed　of　gravitating　and　rotating　matter；　and　because　its　shape　results　from　the　forces　to　which　it　is　subject。　　It　is　unlike　a　star　in　that　it　possesses　the　attributes　of　incompressibility　and　of　uniform　density。　　The　difference　between　the　real　and　the　ideal　is　doubtless　great；　yet　the　similarity　is　great　enough　to　allow　us　to　extend　many　of　the　conclusions　as　to　ideal　liquid　stars　to　the　conditions　which　must　hold　good　in　reality。　　Thus　with　the　object　of　obtaining　some　insight　into　actuality；　it　is　justifiable　to　discuss　an　avowedly　ideal　problem　at　some　length。

The　attraction　of　gravity　alone　tends　to　make　a　mass　of　liquid　assume　the　shape　of　a　sphere；　and　the　effects　of　rotation；　summarised　under　the　name　of　centrifugal　force；　are　such　that　the　liquid　seeks　to　spread　itself　outwards　from　the　axis　of　rotation。　　It　is　a　singular　fact　that　it　is　unnecessary　to　take　any　account　of　the　size　of　the　mass　of　liquid　under　consideration；　because　the　shape　assumed　is　exactly　the　same　whether　the　mass　be　small　or　large；　and　this　renders　the　statement　of　results　much　easier　than　would　otherwise　be　the　case。

A　mass　of　liquid　at　rest　will　obviously　assume　the　shape　of　a　sphere；　under　the　influence　of　gravitation；　and　it　is　a　stable　form；　because　any　oscillation　of　the　liquid　which　might　be　started　would　gradually　die　away　under　the　influence　of　friction；　however　small。　　If　now　we　impart　to　the　whole　mass　of　liquid　a　small　speed　of　rotation　about　some　axis；　which　may　be　called　the　polar　axis；　in　such　a　way　that　there　are　no　internal　currents　and　so　that　it　spins　in　the　same　way　as　if　it　were　solid；　the　shape　will　become　slightly　flattened　like　an　orange。　　Although　the　earth　and　the　other　planets　are　not　homogeneous　they　behave　in　the　same　way；　and　are　flattened　at　the　poles　and　protuberant　at　the　equator。　　This　shape　may　therefore　conveniently　be　described　as　planetary。

If　the　planetary　body　be　slightly　deformed　the　forces　of　restitution　are　slightly　less　than　they　were　for　the　sphere；　the　shape　is　stable　but　somewhat　less　so　than　the　sphere。　　We　have　then　a　planetary　spheroid；　rotating　slowly；　slightly　flattened　at　the　poles；　with　a　high　degree　of　stability；　and　possessing　a　certain　amount　of　rotational　momentum。　　Let　us　suppose　this　ideal　liquid　star　to　be　somewhere　in　stellar　space　far　removed　from　all　other　bodies；　then　it　is　subject　to　no　external　forces；　and　any　change　which　ensues　must　come　from　inside。　　Now　the　amount　of　rotational　momentum　existing　in　a　system　in　motion　can　neither　be　created　nor　destroyed　by　any　internal　causes；　and　therefore；　whatever　happens；　the　amount　of　rotational　momentum　possessed　by　the　star　must　remain　absolutely　constant。

A　real　star　radiates　heat；　and　as　it　cools　it　shrinks。　　Let　us　suppose　then　that　our　ideal　star　also　radiates　and　shrinks；　but　let　the　process　proceed　so　slowly　that　any　internal　currents　generated　in　the　liquid　by　the　cooling　are　annulled　so　quickly　by　fluid　friction　as　to　be　insignificant；　further　let　the　liquid　always　remain　at　any　instant　incompressible　and　homogeneous。　All　that　we　are　concerned　with　is　that；　as　time　passes；　the　liquid　star　shrinks；　rotates　in　one　piece　as　if　it　were　solid；　and　remains　incompressible　and　homogeneous。　　The　condition　is　of　course　artificial；　but　it　represents　the　actual　processes　of　nature　as　well　as　may　be；　consistently　with　the　postulated　incompressibility　and　homogeneity。　　（Mathematicians　are　accustomed　to　regard　the　density　as　constant　and　the　rotational　momentum　as　increasing。　　But　the　way　of　looking　at　the　matter；　which　I　have　adopted；　is　easier　of　comprehension；　and　it　comes　to　the　same　in　the　end。）

The　shrinkage　of　a　constant　mass　of　matter　involves　an　increase　of　its　density；　and　we　have　therefore　to　trace　the　changes　which　supervene　as　the　star　shrinks；　and　as　the　liquid　of　which　it　is　composed　increases　in　density。　　The　shrinkage　will；　in　ordinary　parlance；　bring　the　weights　nearer　to　the　axis　of　rotation。　　Hence　in　order　to　keep　up　the　rotational　momentum；　which　as　we　have　seen　must　remain　constant；　the　mass　must　rotate　quicker。　　The　greater　speed　of　rotation　augments　the　importance　of　centrifugal　force　compared　with　that　of　gravity；　and　as　the　flattening　of　the　planetary　spheroid　was　due　to　centrifugal　force；　that　flattening　is　increased；　in　other　words　the　ellipticity　of　the　planetary　spheroid　increases。

As　the　shrinkage　and　corresponding　increase　of　density　proceed；　the　planetary　spheroid　becomes　more　and　more　elliptic；　and　the　succession　of　forms　constitutes　a　family　classified　according　to　the　density　of　the　liquid。　　The　specific　mark　of　this　family　is　the　flattening　or　ellipticity。

Now　consider　the　stability　of　the　system；　we　have　seen　that　the　spheroid　with　a　slow　rotation；　which　forms　our　starting…point；　was　slightly　less　stable　than　the　sphere；　and　as　we　proceed　through　the　family　of　ever　flatter　ellipsoids　the　stability　continues　to　diminish。　　At　length　when　it　has　assumed　the　shape　shown　in　a　figure　titled　〃Planetary　spheroid　just　becoming　unstable〃　（Fig。　2。）　where　the　equatorial　and　polar　axes　are　proportional　to　the　numbers　1000　and　583；　the　stability　has　just　disappeared。　　According　to　the　general　principle　explained　above　this　is　a　form　of　bifurcation；　and　corresponds　to　the　form　denoted　A。　　The　specific　difference　a　of　this　family　must　be　regarded　as　the　excess　of　the　ellipticity　of　this　figure　above　that　of　all　the　earlier　ones；　beginning　with　the　slightly　flattened　planetary　spheroid。　　Accordingly　the　specific　difference　a　of　the　family　has　gradually　diminished　from　the　beginning　and　vanishes　at　this　stage。

According　to　Poincare's　principle　the　vanishing　of　the　stability　serves　us　with　notice　that　we　have　reached　a　figure　of　bifurcation；　and　it　becomes　necessary　to　inquire　what　is　the　nature　of　the　specific　difference　of　the　new　family　of　figures　which　must　be　coalescent　with　the　old　one　at　this　stage。　　This　difference　is　found　to　reside　in　the　fact　that　the　equator；　which　in　the　planetary　family　has　hitherto　been　circular　in　section；　tends　to　become　elliptic。　　Hitherto　the　rotational　momentum　has　been　kept　up　to　its　constant　value　partly　by　greater　speed　of　rotation　and　partly　by　a　symmetrical　bulging　of　the　equator。　　But　now　while　the　speed　of　rotation　still　increases　（The　mathematician　familiar　with　Jacobi's　ellipsoid　will　find　that　this　is　correct；　although　in　the　usual　mode　of　exposition；　alluded　to　above　in　a　footnote；　the　speed　diminishes。）；　the　equator　tends　to　bulge　outwards　at　two　diametrically　opposite　points　and　to　be　flattened　midway　between　these　protuberances。　　The　specific　difference　in　the　new　family；　denoted　in　the　general　sketch　by　b；　is　this　ellipticity　of　the　equator。　　If　we　had　traced　the　planetary　figures　with　circular　equators　beyond　this　stage　A；　we　should　have　found　them　to　have　become　unstable；　and　the　stability　has　been　shunted　off　along　the　A　＋　b　family　of　forms　with　elliptic　equators。

This　new　series　of　figures；　generally　named　after　the　great　mathematician　Jacobi；　is　at　first　only　just　stable；　but　as　the　density　increases　the　stability　increases；　reaches　a　maximum　and　then　declines。　　As　this　goes　on　the　equator　of　these　Jacobian　figures　becomes　more　and　more　elliptic；　so　that　the　shape　is　considerably　elongated　in　a　direction　at　right　angles　to　the　axis　of　rotation。

At　length　when　the　longest　axis　of　the　three　has　become　about　three　times　as　long　as　the　shortest　（The　three　axes　of　the　ellipsoid　are　then　proportional　to　1000；　432；　343。）；　the　stability　of　this　family　of　figures　vanishes；　and　we　have　reached　a　new　form　of　bifurcation　and　must　look　for　a　new　type　of　figure　along　which　the　stable　development　will　presumably　extend。　　Two　sections　of　this　critical　Jacobian　figure；　which　is　a　figure　of　bifurcation；　are　shown　by　the　dotted　lines　in　a　figure　titled　〃The　'pear…shaped　figure'　and　the　Jocobian　figure　from　which　it　is　derived〃　（Fig。　3。）　comprising　two　figures；　one　above　the　other：　　the　upper　figure　is　the　equatorial　section　at　right　angles　to　the　axis　of　rotation；　the　lower　figure　is　a　section　through　the　axis。

Now　Poincare　has　proved　that　the　new　type　of　figure　is　to　be　derived　from　the　figure　of　bifurcation　by　causing　one　of　the　ends　to　be　prolonged　into　a　snout　and　by　bluntening　the　other　end。　　The　snout　forms　a　sort　of　stalk；　and　between　the　stalk　and　the　axis　of　rotation　the　surface　is　somewhat　flattened。　　These　are　the　characteristics　of　a　pear；　and　the　figure　has　therefore　been　called　the　〃pear…shaped　figure　of　equilibrium。〃　　The　firm　line　shows