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Na pervuyu stranicu << 3.10. Osnovy nebesnoi mehaniki | Oglavlenie | 3.10.2. Parametry i anomalii >>

3.10.1. Zakony Keplera

Dlya opredeleniya sistemy koordinat neobhodimo snachala opredelit' ploskost', zatem v ploskosti opredelit' napravlenie na vydelennuyu tochku. Togda s edinichnym vektorom $ \mathbf{i}$, napravlennym v etu tochku, mozhno svyazat' os' $ x$, s perpendikulyarom k ploskosti -- edinichnyi vektor $ \mathbf{k}$ i os' $ z$; edinichnyi vektor $ \mathbf{j}$ (os' $ y$ sistemy koordinat) opredelyaetsya na osnove vektornogo proizvedeniya tak, chtoby sistema osei byla pravoi ( $ {\mathbf{j}}={\mathbf{k}}\times {\mathbf{i}}$).

Rassmotrim vopros, kak v prostranstve opredelit' etu ploskost' i osi, lezhashie v ploskosti.

V osnove dinamicheskogo metoda opredeleniya sistemy koordinat lezhat uravneniya dinamiki -- i v pervuyu ochered' zakon prityazheniya N'yutona. Soglasno etomu zakonu dva tela s massami $ m_1$ i $ m_2$ prityagivayutsya s siloi $ Gm_1m_2/r^2$, gde $ r$ -- rasstoyanie mezhdu telami. Koefficient $ G=6,673\cdot 10^{-11}\ \textrm{m}^3\textrm{kg}^{-1}\textrm{s}^{-2}$ nazyvaetsya postoyannoi tyagoteniya. Esli tela raspolozheny v tochkah $ P_1$ i $ P_2$ s dekartovymi koordinatami $ x_1,y_1,z_1$ i $ x_2,y_2,z_2$, sootvetstvenno, to dvizhenie tela s massoi $ m_2$ opisyvaetsya uravneniyami:

$\displaystyle m_2{\overset{..}{x}}_2$ $\displaystyle =-Gm_1m_2\frac{x_2-x_1}{r^3},$    
$\displaystyle m_2{\overset{..}{y}}_2$ $\displaystyle =-Gm_1m_2\frac{y_2-y_1}{r^3},$ (30)
$\displaystyle m_2{\overset{..}{z}}_2$ $\displaystyle =-Gm_1m_2\frac{z_2-z_1}{r^3};$    

tochkami oboznacheno differencirovanie po vremeni: $ \overset{..}{x}_2=d^2x_2/dt^2$ i t.d. Pod deistviem toi zhe sily, no protivopolozhnogo znaka, telo s massoi $ m_1$ dvizhetsya soglasno uravneniyam:

$\displaystyle m_1{\overset{..}{x}}_1$ $\displaystyle =Gm_1m_2\frac{x_2-x_1}{r^3},$    
$\displaystyle m_1{\overset{..}{y}}_1$ $\displaystyle =Gm_1m_2\frac{y_2-y_1}{r^3},$ (31)
$\displaystyle m_1{\overset{..}{z}}_1$ $\displaystyle =Gm_1m_2\frac{z_2-z_1}{r^3}.$    

Vvedem oboznacheniya: $ \xi=x_2-x_1$, $ \eta=y_2-y_1$, $ \zeta=z_2-z_1$, $ \mu=G(m_1+m_2)$. Vychitaya iz uravnenii (3.28) uravneniya (3.29), poluchim

$\displaystyle {\overset{..}{\xi}} +\mu\frac{\xi}{r^3}$ $\displaystyle =0,$ (32)
$\displaystyle {\overset{..}{\eta}} +\mu\frac{\eta}{r^3}$ $\displaystyle =0,$ (33)
$\displaystyle {\overset{..}{\zeta}} +\mu\frac{\zeta}{r^3}$ $\displaystyle =0.$ (34)

V uravneniya (3.30-3.32) vhodyat lish' otnositel'nye koordinaty dvuh tochek, t.e. uravneniya dvizheniya ne zavisyat ot polozheniya nachala sistemy koordinat. Umnozhaya uravnenie (3.30) na $ \eta$, a uravnenie (3.31) na $ \xi$, i zatem vychitaya iz pervogo uravneniya vtoroe, poluchim

$\displaystyle \eta {\overset{..}{\xi}} - \xi{\overset{..}{\eta}}=0 $

ili

$\displaystyle \frac{d}{dt}(\eta\overset{.}{\xi}-\xi\overset{.}{\eta})=0.$ (3.35)

Iz (3.33) sleduet, chto velichina v skobkah ne zavisit ot vremeni, t.e.

$\displaystyle \eta\overset{.}{\xi}-\xi\overset{.}{\eta}=A=\textrm{const}.$ (3.36)

Analogichnym obrazom iz uravnenii (3.31),(3.32) poluchim vyrazhenie:

$\displaystyle \zeta\overset{.}{\eta}-\overset{.}{\zeta} \eta=B=\textrm{const},$ (3.37)

a iz (3.30) i (3.32):

$\displaystyle -\zeta\overset{.}{\xi}+\overset{.}{\zeta} \xi=C=\textrm{const}.$ (3.38)

Uravneniya (3.34-3.36) nazyvayutsya integralami ploshadei, a postoyannye $ A,B,C$ -- postoyannymi ploshadei.

Umnozhaya uravnenie (3.34) na $ \zeta$, (3.35) -- na $ \xi$, (3.36) -- na $ \eta$ i skladyvaya, nahodim, chto

$\displaystyle A\zeta +B\xi +C\eta=0.$ (3.39)

Uravnenie (3.37) -- eto uravnenie ploskosti. Znachit, dva tela, dvizhushiesya v prostranstve pod deistviem sily prityazheniya, vsegda nahodyatsya v odnoi i toi zhe ploskosti; traektoriya tela 2 otnositel'no tela 1 yavlyaetsya ploskoi krivoi i nazyvaetsya orbitoi. Drugimi slovami orbita odnogo tela otnositel'no drugogo lezhit v ploskosti.

Raspolozhim osi $ Ox,Oy$ sistemy koordinat, kotoruyu my hotim opredelit' v ploskosti orbity, a os' $ Oz$ budet perpendikulyarna ei. Tochku $ O$ (nachalo sistemy koordinat) sovmestim s telom s massoi $ m_1$. Togda uravneniya dvizheniya (3.30-3.32) mozhno zapisat' v vide:

$\displaystyle \overset{..}{\mathbf{r}}+\mu\frac{\mathbf{r}}{r^3}=0,$ (3.40)

gde $ \mathbf{r}$ -- vektor, napravlennyi ot tela 1 k telu 2. Umnozhaya vektorno (3.38) na $ \mathbf{r}$ sleva, poluchim

$\displaystyle \mathbf{r}\times \frac{d^2\mathbf{r}}{dt^2}=0, $

tak kak $ \mathbf{r}\times\mathbf{r}=0$. Integriruya, my poluchim, chto

$\displaystyle \mathbf{r}\times\frac{d\mathbf{r}}{dt}=\mathbf{h},$ (3.41)

gde $ \mathbf{h}$--ne zavisyashii ot vremeni vektor. Vektor $ \mathbf{h}$ nazyvaetsya vektorom uglovogo momenta, i soglasno opredeleniyu vektornogo proizvedeniya on perpendikulyaren ploskosti orbity, v kotoroi lezhat i radius-vektor $ \mathbf{r}$, i vektor skorosti $ d\mathbf{r}/dt$. Uravnenie (3.39) ekvivalentno uravneniyu (3.37): postoyannye $ A,B,C$ predstavlyayut soboi proekcii $ \mathbf{h}$ na osi inercial'noi sistemy koordinat.

Umnozhim teper' uravneniya (3.30-3.32) sootvetstvenno na $ 2\overset{.}{\xi}$, $ 2\overset{.}{\eta}$, $ 2\overset{.}{\zeta}$ i slozhim. Koordinaty $ \xi,\eta,\zeta$ yavlyayutsya koordinatami tela 2 otnositel'no tela 1. V rezul'tate poluchim sleduyushee uravnenie:

$\displaystyle 2\overset{.}{\xi}{\overset{..}{\xi}}+2\overset{.}{\eta}{\overset{..}{\eta}}+2\overset{.}{\zeta}{\overset{..}{\zeta}} = -\frac{2\mu}{r^3}(\xi\overset{.}{\xi}+\eta\overset{.}{\eta}+\zeta\overset{.}{\zeta}). $

Tak kak $ r^2=\xi^2+\eta^2+\zeta^2$, to imeem

$\displaystyle \xi\overset{.}{\xi}+\eta\overset{.}{\eta}+\zeta\overset{.}{\zeta} = r{\overset{.}{r}}, $

vsledstvie chego predydushee uravnenie primet vid

$\displaystyle \frac{d}{dt}({\overset{.}{\xi}}^2+{\overset{.}{\eta}}^2+{\overset{.}{\zeta}}^2) = -\frac{2\mu}{r^2}{\overset{.}{r}}= \frac{d}{dt}\left(\frac{2\mu}{r}\right). $

Integrirovanie uravneniya daet:

$\displaystyle V^2= \frac{2\mu}{r}+W,$ (3.42)

gde $ V^2={\overset{.}{\xi}}^2+{\overset{.}{\eta}}^2+{\overset{.}{\zeta}}^2$ -- kvadrat skorosti tela 2, dvizhushegosya otnositel'no tela 1. Proizvol'naya postoyannaya $ W$ v uravnenii (3.40) nazyvaetsya postoyannoi energii. Ona mozhet byt' velichinoi ravnoi nulyu, polozhitel'noi ili otricatel'noi. V nebesnoi mehanike dokazyvaetsya, chto ot velichiny postoyannoi energii zavisit tip orbity tela: pri $ W\lt 0$ orbita est' ellips, pri $ W=0$ -- parabola i pri $ W\gt 0$ -- giperbola.

Tak kak orbita lezhit v ploskosti, i polozhenie tela 2 otnositel'no tela 1 opredelyaetsya lish' koordinatami $ x,y$, to udobno dlya dal'neishih vychislenii vvesti polyarnye koordinaty $ r,\theta$ (ris. 3.12), tak chto

$\displaystyle x$ $\displaystyle =r\cos\theta,$    
$\displaystyle y$ $\displaystyle =r\sin\theta.$    

Ris. 3.12. Opredelenie polyarnyh koordinat

V polyarnoi sisteme koordinat vvedem dva edinichnyh vektora $ \mathbf{i}_r,\mathbf{i}_{\theta}$, prichem pervyi iz nih napravlen vdol' $ \mathbf{r}$, a vtoroi--perpendikulyaren emu i napravlen v storonu uvelicheniya ugla $ \theta$. Togda

$\displaystyle \overset{.}{x}$ $\displaystyle =\frac{dx}{dt}=\overset{.}{r}\cos\theta -r\sin\theta \overset{.}{\theta},$    
$\displaystyle \overset{.}{y}$ $\displaystyle =\frac{dy}{dt}=\overset{.}{r}\sin\theta +r\cos\theta \overset{.}{\theta}$    

ili v vektornom vide

$\displaystyle \frac{d\mathbf{r}}{dt}=\mathbf{i}_r\overset{.}{r} +\mathbf{i}_{\theta}r\overset{.}{\theta}.$ (3.43)

Sledovatel'no, v polyarnyh koordinatah uravnenie uglovogo momenta (3.39) imeet vid:

$\displaystyle \mathbf{i}_r r\times(\mathbf{i}_r\overset{.}{r} +\mathbf{i}_{\theta}r\overset{.}{\theta})=\mathbf{h} $

ili

$\displaystyle \mathbf{i}_r\times\mathbf{i}_{\theta}r^2\overset{.}{\theta}=\mathbf{k}r^2\overset{.}{\theta}=\mathbf{k}h, $

gde $ \mathbf{k}$--edinichnyi vektor, napravlennyi vdol' vektora $ \mathbf{h}$ i, soglasno nashemu opredeleniyu sovpadayushii s napravleniem osi $ Oz$. Znachit,

$\displaystyle r^2\frac{d\theta}{dt}=h.$ (3.44)

Dopustim, chto v moment $ t$ telo 2 nahodilos' na rasstoyanii $ r$ ot tela 1, a cherez promezhutok vremeni $ \Delta t$ peremestilos' na ugol $ \Delta\theta$, prichem rasstoyanie stalo ravnyat'sya $ r+\Delta r$. Schitaya, chto promezhutok vremeni $ \Delta t$ mal, mozhno schitat' dugu, po kotoroi dvizhetsya telo 2, pryamoi liniei. Togda ploshad' sektora, kotoryi obrazuyut dva radius-vektora $ r$ i $ r+\Delta r$, budet blizok k ploshadi treugol'nika, ravnoi $ \frac{1}{2}r(r+\Delta r)\sin\Delta\theta$. Ustremlyaya $ \Delta\theta$ k nulyu i delya na promezhutok vremeni $ \Delta t\rightarrow 0$, nahodim, chto ploshad' sektora, opisyvaemaya telom ravna $ \frac{1}{2}r^2\frac{d\theta}{dt}$. Sledovatel'no, na osnove uravneniya (3.42) mozhno utverzhdat', chto za odinakovye promezhutki vremeni radius-vektor opisyvaet ravnye ploshadi, prichem velichina uglovogo momenta ravna udvoennoi ploshadi sektora. Eto -- vtoroi zakon Keplera.

Zapishem teper' uravnenie (3.38) v polyarnyh koordinatah. Tak kak proizvodnaya $ \overset{.}{\mathbf{r}}$ uzhe naidena (3.41), to

$\displaystyle \overset{..}{\mathbf{r}}=\overset{..}{r} {\mathbf{i}}_r+\overset{.}{r}\frac{d{\mathbf{i}}_r}{dt}+ \frac{d(r\overset{.}{\theta})}{dt}{\mathbf{i}}_\theta+r\overset{.}{\theta}\frac{d{\mathbf{i}}_\theta}{dt}.$ (3.45)

Edinichnye vektory $ {\mathbf{i}}_r$, $ {\mathbf{i}}_{\theta}$ menyayut napravlenie so vremenem, poetomu menyayutsya ih proekcii na osi $ x,y$. Sledovatel'no, proizvodnye $ d{\mathbf{i}}_r/dt$, $ d{\mathbf{i}}_{\theta}/dt$ ne ravny nulyu. Chtoby ih vychislit', naidem proizvodnuyu edinichnogo vektora $ {\mathbf{i}}_r$ po uglu $ \theta$ (ris. 3.12). Tak kak

$\displaystyle {\mathbf{i}}_r={\mathbf{i}}\cos\theta+{\mathbf{j}}\sin\theta, $

to,

$\displaystyle \frac{d{\mathbf{i}}_r}{d\theta} =-{\mathbf{i}}\sin\theta+{\mathbf{j}}\cos\theta={\mathbf{i}}_r(\theta+\frac{\pi}{2})={\mathbf{i}}_{\theta}. $

Iz poslednego vyrazheniya nahodim

$\displaystyle \frac{d{\mathbf{i}}_r}{dt}= \frac{d{\mathbf{i}}_r}{d\theta}\frac{d\theta}{dt}={\mathbf{i}}_{\theta}\overset{.}{\theta}. $

Analogichnym obrazom naidem vyrazhenie dlya proizvodnoi $ {d{\mathbf{i}}_\theta}/dt$:

$\displaystyle \frac{d{\mathbf{i}}_\theta}{d\theta} =-{\mathbf{i}}\cos\theta-{\mathbf{j}}\sin\theta=-{\mathbf{i}}_r, \quad \frac{d{\mathbf{i}}_\theta}{dt}= \frac{d{\mathbf{i}}_\theta}{d\theta}\frac{d\theta}{dt}=-{\mathbf{i}}_r\overset{.}{\theta}. $

Podstavlyaya znacheniya proizvodnyh $ {d{\mathbf{i}}_r}/dt$, $ {d{\mathbf{i}}_\theta}/dt$ v uravnenie (3.43) i privodya podobnye chleny, poluchim, chto uskorenie tela razlagaetsya na dve komponenty -- radial'nuyu i normal'nuyu sostavlyayushie:

$\displaystyle \overset{..}{\mathbf{r}}=(\overset{..}{r}-r\overset{.}{\theta}^2) {\mathbf{i}}_r+ (r\overset{..}{\theta} +2\overset{.}{r}\overset{.}{\theta}) {\mathbf{i}}_\theta. $

Tak kak vtoroi chlen v skobkah mozhno zapisat' v vide:

$\displaystyle r\overset{..}{\theta} +2\overset{.}{r}\overset{.}{\theta} = \frac{1}{r}\frac{d}{dt} \Bigl(r^2\frac{d\theta}{dt}\Bigr), $

to iz vtorogo zakona Keplera (3.42) sleduet ego ravenstvo (normal'noi sostavlyayushei uskoreniya) nulyu.

Polagaya, chto $ {\mathbf{r}}={\mathbf{i}}_r r$, zapishem uravnenie (3.38) v polyarnyh koordinatah v sleduyushem vide:

$\displaystyle \overset{..}{r} - r \overset{.}{\theta}^2 =-\frac{\mu}{r^2}.$ (3.46)

Differencial'nye uravneniya (3.42) i (3.44) opisyvayut zavisimost' rasstoyaniya $ r$ odnogo tela otnositel'no drugogo i ugla $ \theta$ ot vremeni. Dlya resheniya etih uravnenii obychno isklyuchayut vremya iz (3.44) s pomosh'yu (3.42). Dlya udobstva vvedem parametr $ u$, tak chto

$\displaystyle u=\frac{1}{r}. $

Togda zakon Keplera (3.42) zapisyvaetsya v vide: $ \overset{.}{\theta}=hu^2$. Teper' vyrazim proizvodnuyu $ \overset{..}{r}$ cherez parametr $ u$. Dlya etogo naidem snachala proizvodnuyu $ \overset{.}{r}$:

$\displaystyle \overset{.}{r}=\frac{d}{dt}\Bigl(\frac{1}{u}\Bigr)= -\frac{1}{u^2}\frac{du}{dt} = -\frac{1}{u^2}\frac{du}{d\theta}\frac{d\theta}{dt}=-h\frac{du}{d\theta}, $

i, uchityvaya, chto $ \overset{.}{r}$ yavlyaetsya neyavnoi funkciei $ \theta$, $ h=\textrm{const}$, poluchim

$\displaystyle \overset{..}{r}=\frac{d}{dt}(\overset{.}{r}) = \frac{d\overset{.}{r}}{d\theta}\frac{d\theta}{dt}=-h^2u^2\frac{d^2u}{d\theta^2}. $

Posle podstanovki $ \overset{..}{r}$ v uravnenie (3.44) naidem:

$\displaystyle -h^2u^2\frac{d^2u}{d\theta^2}-h^2u^3=-\mu u^2 $

ili

$\displaystyle \frac{d^2u}{d\theta^2}+u=\frac{\mu}{h^2}.$ (3.47)

Reshenie differencial'nogo uravneniya vtorogo poryadka (3.45) zapisyvaetsya v vide:

$\displaystyle u=\frac{\mu}{h^2} + A\cos(\theta-\omega), $

gde $ A$ i $ \omega$ -- dve konstanty integrirovaniya. Neposredstvennoi podstanovkoi mozhno ubedit'sya, chto $ u$ yavlyaetsya resheniem uravneniya (3.45). Zamenyaya $ u$ na $ r$ i vvodya novye parametry: $ p=h^2/\mu$, $ e=Ah^2/\mu$, nahodim uravnenie traektorii tela v polyarnyh koordinatah:

$\displaystyle r=\frac{p}{1+e\cos(\theta-\omega)}.$ (3.48)

Uravnenie (3.46) yavlyaetsya uravneniem konicheskih sechenii. Vid orbity zavisit ot parametra $ e$ -- ekscentrisiteta orbity. Esli $ 0\leq e \lt 1$, to traektoriya yavlyaetsya ellipsom, esli $ e=1$, to -- paraboloi, esli $ e\gt 1$, to -- giperboloi. Vid orbity mozhno opredelit' takzhe po velichine postoyannoi energii v uravnenii (3.40), kotoraya zavisit ot skorosti i radiusa-vektora tela. Poetomu udobno svyazat' vid orbity s nachal'nymi parametrami $ V_0$ i $ r_0$:

$\displaystyle \textrm{esli}\ V_0^2\lt \frac{2\mu}{r_0},\ \textrm{to}\ W\lt 0\ \textrm{i}\ 0\leq e\lt 1\ \textrm{ - ellipticheskaya orbita},$    
$\displaystyle \textrm{esli}\ V_0^2= \frac{2\mu}{r_0},\ \textrm{to}\ W=0\ \textrm{i}\ e=1\ \textrm{ - parabola},$ (49)
$\displaystyle \textrm{esli}\ V_0^2\gt \frac{2\mu}{r_0},\ \textrm{to}\ W\gt 0\ \textrm{i}\ e\gt 1\ \textrm{ - giperbolicheskaya orbita}.$    

Ogranichimsya seichas sluchaem, kogda $ 0\leq e \lt 1$. V etom sluchae uravnenie (3.46) yavlyaetsya matematicheskoi formoi pervogo zakona Keplera.

Esli telo s massoi $ m_1$ nazvat' Solncem, drugoe telo -- planetoi, to pervyi zakon Keplera formuliruetsya sleduyushim obrazom: planeta dvizhetsya po ellipsu, v odnom iz fokusov kotorogo nahoditsya Solnce. Parametr $ p$ nazyvaetsya parametrom ellipsa i svyazan s bol'shoi poluos'yu $ a$ ellipsa formuloi: $ p=a(1-e^2)$. Malaya poluos' $ b$ mozhet byt' vyrazhena cherez $ a$ i $ e$: $ b^2=a^2(1-e^2)$ (ris. 3.13). Na ris. 3.13 Solnce nahoditsya v tochke $ O$, planeta -- v tochke $ P$, os' $ OX$ napravlena v tochku voshodyashego uzla orbity, a os' $ Ox$ -- v tochku orbity, blizhaishei k Solncu, kotoraya nazyvaetsya perigeliem. Ugol $ \omega$ nazyvaetsya dolgotoi perigeliya.

Ris. 3.13. Opredelenie parametrov ellipsa

Esli oboznachit' period obrasheniya planety $ P$ kak $ T$, to soglasno vtoromu zakonu Keplera za vremya $ T$ planeta opishet polnyi ellips, ploshad' kotorogo ravna $ \pi ab$. Otnoshenie ploshadi ellipsa k periodu obrasheniya ravno polovine uglovogo momenta planety, t.e.

$\displaystyle \frac{\pi ab}{T}=\frac{h}{2}=\textrm{const}. $

Sledovatel'no

$\displaystyle 2\pi a^2\sqrt{1-e^2}=hT.$ (3.50)

Tak kak $ h^2/\mu=p=a(1-e^2)$, to $ h^2=\mu a(1-e^2)$. Isklyuchaya $ h$ iz (3.48), nahodim:

$\displaystyle T=2\pi\sqrt{\frac{a^3}{\mu}}. $

Period obrasheniya zavisit tol'ko ot summy mass tel, tak kak $ \mu=G(m_1+m_2)$, i velichiny bol'shoi poluosi orbity.

V sluchae, kogda na tela 1 i 2 ne deistvuyut sily prityazheniya drugih tel (v nebesnoi mehanike eta zadacha tak i nazyvaetsya zadachei dvuh tel), period obrasheniya est' velichina postoyannaya i mozhet sluzhit' edinicei vremeni. V nachale XX veka na osnove nablyudenii Solnca i Luny formirovalas' shkala efemeridnogo vremeni (Ephemeris Time, ET). Tak kak iz-za vozmushenii orbity drugimi telami period obrasheniya menyaetsya, dlya postroeniya shkaly ET neobhodimy byli dlitel'nye nablyudeniya. Iz-za slozhnosti postroeniya etoi shkaly, a takzhe iz-za poyavleniya v seredine 50-h godov XX veka atomnyh standartov chastoty ot shkaly vremeni, osnovannoi na obrashenii Zemli vokrug Solnca, prishlos' otkazat'sya. V nastoyashee vremya v osnove scheta vremeni lezhit atomnaya shkala vremeni TAI, odnako samoi stabil'noi na bol'shih intervalah vremeni mozhet okazat'sya pul'sarnaya shkala vremeni, prichem pul'sar yavlyaetsya odnoi iz zvezd v dvoinoi sisteme.

Oboznachim cherez $ n$ srednyuyu skorost' dvizheniya planety:

$\displaystyle n=\frac{2\pi}{T}=\sqrt{\frac{\mu}{a^3}}.$ (3.51)

V nebesnoi mehanike parametr $ n$ nazyvaetsya srednim dvizheniem. Esli massu Solnca oboznachit' kak $ M_\odot$, massu planety -- kak $ M_{P_1}$, prichem period obrasheniya i bol'shaya poluos' ravny $ T_1$ i $ a_1$, to

$\displaystyle G(M_\odot +M_{P_1}) =\frac{4\pi^2a_1^3}{T_1^2}=n_1^2a_1^3.$ (3.52)

Analogichnoe uravnenie mozhno napisat' dlya drugoi planety s massoi $ M_{P_2}$, periodom obrasheniya $ T_2$ i bol'shoi poluos'yu $ a_1$:

$\displaystyle G(M_\odot +M_{P_2}) =\frac{4\pi^2a_2^3}{T_2^2}=n_2^2a_2^3. $

Delya odno uravnenie na drugoe, poluchim:

$\displaystyle \frac{G(M_\odot +M_{P_1})}{G(M_\odot +M_{P_2})}= \Bigl(\frac{a_1}{a_2}\Bigr)^3\Bigl(\frac{T_2}{T_1}\Bigr)^2 = \Bigl(\frac{a_1}{a_2}\Bigr)^3\Bigl(\frac{n_1}{n_2}\Bigr)^2.$ (3.53)

Uravnenie (3.51) yavlyaetsya matematicheskoi zapis'yu tret'ego zakona Keplera. Tak kak dlya samoi massivnoi planety v solnechnoi sisteme -- Yupitera otnoshenie $ M_P/M_\odot$ $ \sim 10^{-3}$, to velichina v levoi chasti (3.51) otlichaetsya ot edinicy v tret'em znake. Sledovatel'no, s tochnost'yu do $ 10^{-3}$ imeem

$\displaystyle \Bigl(\frac{a_1}{a_2}\Bigr)^3=\Bigl(\frac{T_1}{T_2}\Bigr)^2.$ (3.54)

Kvadraty periodov obrasheniya planet otnosyatsya kak kuby ih bol'shih poluosei. Opredelyaya bol'shuyu poluos' $ a_1$ dlya Zemli kak 1 astronomicheskuyu edinicu (1 a.e.) (eto -- nepravil'noe opredelenie astronomicheskoi edinicy; sm. glavu 8) i $ T_1$ kak 1 god, to, izmeryaya period obrasheniya kakoi libo planety (v godah), mozhno zapisat' tretii zakon Keplera v forme:

$\displaystyle a^3=T^2, $

gde $ a,T$ -- bol'shaya poluos' i period obrasheniya lyuboi planety. Takim obrazom tretii zakon Keplera ustanavlivaet lish' otnositel'nye razmery orbit planet. Chtoby ustanovit' istinnye razmery v solnechnoi sisteme neobhodimo znat' velichinu 1 a.e. v metrah. V nachale XX veka dlya etogo ispol'zovalis' nablyudeniya Solnca i vychislyalas' velichina solnechnogo parallaksa (sm. razdel 6.3.2). Zatem na smenu opticheskim nablyudeniyam prishli bolee tochnye metody radiolokacii planet, chto pozvolilo opredelit' znachenie 1 a.e. s oshibkoi v neskol'ko metrov.


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