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$BC+@n@6N4(B $BLu(B
\end{flushright}

\noindent
{\it Nature} {\bf 386} (1997), 254-256

\vspace{0.3cm}
\begin{center}
{\large
{\bf 16 Cygni B$B$r2s$kOG@150F;$NN%?4N($N%+%*%9E*JQF0(B \\
Chaotic variations in the eccentricity of \\
the planet orbiting 16 Cygni B}
}

\vspace{0.3cm}
Matthew Holman, Jihad Touma \& Scott Tremaine
\end{center}

\vspace{0.5cm}
\noindent
{\bf $BMWLs(B.} $B:G6a(B16 Cyg B$B$N$^$o$j$r2s$C$F$$$kOG@1(B$^1$$B$,H/8+$5$l$?$,!"(B $B$3$l$OCN$i$l$F(B
$B$$$kOG@1$N$&$A:GBg$NN%?4N((B($e=0.67$)$B$r;}$D(B. 2$B8D0J>e$NOG@1$,$"$k>l9g$K$O(B, 
$B=ENOAj8_:nMQ$K$h$C$F50F;$,Bg$-$JN%?4N($r3MF@$9$k2DG=@-$,$J$$$G$O$J$$$,(B$^{2,3}$,
$B@1$r<h$j4,$/1_HW$+$i$G$-$kOG@1$O$[$\1_50F;$r;}$D$H4|BT$5$l$k(B.
$B$3$NO@J8$G$O(B16 Cyg Bb$B$NN%?4N($NBg$-$J50F;$,1s$$(B
$BH<@1(B16 Cyg A$B$H$N=ENOAj8_:nMQ$+$i@8$8$k$3$H$r<(:6$9$k(B. 16 Cyg Bb$B$,(B16 Cyg A$B$N(B
$B50F;LL$+$i(B$45^{\circ}$$B$J$$$7(B$135^{\circ}$$B79$$$?50F;LL$G$[$\1_50F;>e$K7A@.(B
$B$7$?$H2>Dj$7(B, $B$^$?(B30$BE7J8C10L0JFb$KLZ@1DxEY$N<ANL$NOG@1$,$J$$$H$9$k$H(B, 
16 Cyg Bb$B$O>.N%?4N($HBgN%?4N($N4V$r?6F0$9$k(B. $B$3$l$i$N50F;4V$N0\9T$O(B
$10^7 - 10^9$$BG/$4$H$K5/$3$j(B, $BOG@1$O@836$N(B35 \%$B$r(B$e > 0.6$$B$G2a$4$9(B. 
$B$3$l$i$N7k2L$+$i$9$k$H(B, $BO"@17O$K$$$kOG@1$N>l9g(B, 
$B$?$$$F$$N%?4N($NBg$-$J4|4V$d%+%*%9$N4|4V$r;}$A(B, $B$^$?<g@1$H>WFM$9$k$3$H$b$"$k(B. 

\vspace{0.3cm}
$BOG@1$N1?F0J}Dx<0$N?tCM@QJ,$K$h$k$HN%?4N($H50F;79<P3Q$O?6F0$9$k(B($B?^(B1).
$BC;<~4|6J@~$OO"@1$N6a@1E@5wN%(B$q_b = 100 AU$$B$N>l9g$G$"$j(B, $BD9<~4|6J@~$O(B
$q_b = 500 AU$$B$N>l9g$rI=$9(B. $B?6F0<~4|$K$o$?$C$FN%?4N($,3j$i$+$K%<%m$+$i(B
$e=0.75$$B$^$GA}2C$7(B, $B$^$?$b$H$KLa$k(B. $B5U$K50F;79<P3Q$O(B$i=40^{\circ}$$B$^$G8:>/$7(B, 
$B$^$?$b$H$KLa$k(B. $B?6F0$NI}$O?t<~4|$N4V$K>/$7$7$+JQ$o$i$:(B, $B$I$A$i$N>l9g$b$[$\(B
$B0lDj$G$"$k(B. 16 Cyg Bb$B$O8=:_N%?4N($NBg$-$J0LAj$K$$$k$H;W$o$l$k(B. 

\vspace{4cm}
\hspace*{6cm} $B?^(B1

%\vspace{5cm}
\newpage
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$BO"@1$N50F;<~4|$KHf$Y$F$f$C$/$j$G$"$k(B. $B$@$+$i%,%&%9(B$^4$$B$K$7$?$,$C$FH<@1$r(B
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$e_0 = 0.001$$B$N(B1000$B8D$N!VOG@1!W$N$U$k$^$$$r3QEYMWAG(B($B50F;79<P3Q(B, $B6aE@0z?t(B, 
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$B50F;79<P3Q(B$i_0$$B$N4X?t$H$7$FIA$$$?$b$N$G$"$k(B. $i_0 < 39^{\circ}$$B$N$H$-(B, 
$B:GBgN%?4N($O>.$5$$$^$^$G$"$k(B. $i_0 = 39^{\circ}$$B$GFMA3JQ2=$7(B, 
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$B@836$N(B35 \%$B$r(B$e > 0.6$$B$G2a$4$7F@$k$3$H$r<($7$F$$$k(B. 

\vspace{4cm}
\hspace*{6cm} $B?^(B2

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$B%]%F%s%7%c%k$O<!<0$GM?$($i$l$k(B. 
$$
  \Phi = \frac{GM_s}{4[a_bq_b(1+e_b)]^{3/2}}(2z^2 - x^2 - y^2)
[1 + O(ae_b/q_b)]   \eqno(1)
$$ 
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$B$3$N6a;w$G$O(B, $\Theta = (1-e^2) \cos^2 i$$B$O1?F0$N@QJ,$G$"$k(B(Kozai$B@QJ,(B). 
$B$3$N%b%G%k$G$O(B1$B<+M3EY$7$+;D$i$:(B, $BAj6u4V$N50@W$O(B$\Phi$$B$N%3%s%H%"$K1h$C$FF0$/(B. 
$B$3$l$i$r$&$^$/5-=R$9$k$K$OL5<!851?F0NLJQ?t(B$x = 1-e^2$$B$H$=$l$K6&Lr$J3QEY(B
$B$D$^$j6aE@0z?t(B$\omega$$B$r;H$($P$h$$(B. 
$B?^(B3$B$K$O(B$\Phi = 0.25$($e=0$$B$J$i(B$i = 60^{\circ}$)$B$N>l9g$N(B
$B%3%s%H%"$,<($5$l$F$$$k(B. $BAj6u4V$O(B, $B6aE@0z?t(B$\omega$$B$,=[4D(B(circulate)$B$9$kNN0h$H(B
$\pi/2$$B$N$^$o$j$rGiF0$9$k$KJ,$+$l$k(B. $x=1(e=0)$$B$+$D(B$\omega=0$$B$+$i=PH/$9$k(B
$BOG@1$r9M$($h$&(B. $B;~4V$,7P$D$HOG@1$NN%?4N($O6/@)E*$KBg$-$/$J$C$F(B$\omega=\pi/2$
$B$N$H$-$K:GBg$K$J$j(B, $B$=$N8e$b$H$NCM$KLa$k(B. $\Theta$$B$OJ]B8$9$k$+$i(B$e$$B$,Bg$-$/(B
$B$J$k$H(B$i$$B$O>.$5$/$J$j(B, $B$=$N5U$b@.$jN)$D(B. $B?^(B1$B$G4Q;!$G$-$k$N$,$3$l$G$"$k(B. 
$\Theta$$B$,A}2C$9$k$H(B($B=i4|50F;79<P3Q$,8:>/$9$k$H(B), $BGiF0E@$O>e$K0\F0$7(B, $BGiF0$N(B
$BEg$O>.$5$/$J$k(B. $\Theta \geq 3/5$$B$G$O>CLG$9$k(B. $B$9$k$H50@W$OBg$-$JN%?4N($r(B
$B;}$?$J$/$J$k(B. $e=0$$B$G$O(B$\Theta = 3/5$$B$O(B
$i = \cos^{-1}(3/4)^{1/2} = 39^{\circ}.2$$B$KBP1~$7(B, $B?^(B2$B$NA+0\E@$K0lCW$9$k(B. 


\vspace{4cm}
\hspace*{6cm} $B?^(B3

\vspace{4cm}
$BJ?6Q2=%b%G%k$N2>Dj$r4K$a$k$H!"(B $B50@W$O%3%s%H%"$+$i$5$^$h$$=P$k$3$H$,$G$-$k(B.
$B?^(B3$B$N;6$i$P$C$?E@$O?^(B1$B$N(B$q_b = 100 AU$$B$N?tCM@QJ,$+$i$N%G!<%?$G$"$k(B. 
$BJ?6QA`:n$O$7$F$*$i$:(B, $\Phi$$B$N9`$GMn$H$7$?$N$O(B$O(ae_b/q_b) \approx 2 \%$
$B$J$N$K(B, $BE@$O%3%s%H%"$+$i$O$:$l$J$$(B. $B%;%Q%i%H%j%C%/%9$N6a$/$G50@W$O(B
circulation$B$HGiF0$N4V$r9T$C$?$jMh$?$j$7F@$k(B. $B$3$N8rBX$,%+%*%9$N>I>u$G$"$k(B. 
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$B$N6a$/$KMh$k(B. $B$@$+$i(B16 Cyg Bb$B$N50F;$d;w$?$h$&$JO"@17O$G@8$^$l$?OG@1$N50F;$O(B
$B%+%*%9E*$G$"$m$&(B. 

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$B3d9g$r8+@Q$b$k$3$H$,$G$-$k(B. $\omega=0$$B$K$*$1$k=i4|N%?4N($H50F;79<P3Q$,(B
$BM?$($i$l$k$H(B, $\omega =\pi/2$$B$N$H$-$K(B$e$$B$r7W;;$7$F(B(1)$B$,F@$i$l(B, $B?6F0<~4|$N4V$K(B
$e > 0.6$$B$G2a$4$9;~4V$r@QJ,$7$F(B(2)$B$,F@$i$l$k(B. $B?^(B2$B$N<B@~$HGK@~$O=i4|N%?4N((B
$e_0 = 0.001, 0.01, 0.1$$B$N>l9g$N$3$N7k2L$r<($9(B. $B=i4|50F;79<P3Q(B
$i_0 > 39^{\circ}.2$$B$N$H$-$O:GBgN%?4N($O(B$e_0$$B$K$"$^$j0M$i$:(B, 
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$e_0$$B$K1TIR$K0MB8$9$k(B. $B<B:]>e$O(B, $BJ}Dx<0(B(1)$B$N(B$O(ae_b/q_b)$$B$N9`$ON%?4N($K>.$5$J(B
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Mazeh et al.$^{11}$$B$H(BInnanen et al.$^{12}$$B$,5DO@$7$F$$$k(B.) 

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$BOG@1$N50F;H>D97B$OO"@1$N$b$N$KHf$Y$F$:$C$H>.$5$$$H$9$k$H(B, 
$B$3$N$h$&$JOG@1$r4QB,$9$k3NN($O<!<0$GM?$($i$l$k(B.   
$$
  P(e > e^*) = \int_0^{\pi/2} f	(e^*, e_0, i_0) \sin i_0 d i_0 .  \eqno(2)
$$ 
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$e_0$$B$,(B$0.001$$B$H(B$0.2$$B$N4V$K$"$k$H$9$k$H(B, $BE,Ev$JO"@17O$N(B
10 \%$B$J$$$7(B25 \%$B$NOG@1$O(B$e > 0.6$$B$N$H$-$K4QB,$5$l$k$3$H$,$o$+$k(B. 

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3$B$D$N>r7o$K0MB8$9$k(B. 
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$B50F;H>D97B$,(B40 AU$B$r1[$;$P(B, $BO"@1$N50F;LL$HB7$o$J$$$+$i$G$"$k(B($B;29MJ88%(B7).  
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$BAj4X$,$"$k(B$^8$$B$,4QB,$NA*Br8z2L$K$h$C$F(B$\leq 200yr(a \leq 34 AU)$$B$N<~4|$N$b$N(B
$B$7$+4QB,$5$l$J$$(B. $B$3$l$b(B16 Cyg$B$NCM$h$j$:$C$H>.$5$$(B. $BBhFs$K(B, $BN%?4N(?6F0$N(B
$BH><~4|$OO"@1$NG/Np$h$j>.$5$$$O$:$G$"$k(B. $B?6F0<~4|$O(BKozai$B@QJ,(B$\Theta$$B$*$h$S(B
$BAjJ?LLFb$G$N0LCV$K0MB8$7(B, $B$*$h$=<!<0$GM?$($i$l$k(B. 
$$ \begin{array}{rl}
  P_{precess} & \displaystyle 
\approx P_{orb} \left( \frac{M_p}{M_s} \right) 
\left( \frac{a_b}{a} \right)^3 (1 - e_b^2)^{3/2} \\
 & \displaystyle 
\approx 10^8 yr \left( \frac{P_{orb}}{1 yr} \right) 
\left( \frac{a_b}{1000 AU} \right)^{3/2} 
\left( \frac{q_b}{100 AU} \right)^{3/2} 
\left( \frac{1 AU}{a} \right)^3 .
   \end{array}     \eqno(3)
$$
$B$3$3$G(B$P_{orb}$$B$OOG@1$N50F;<~4|(B, $M_p$$B$O<g@1$N<ANL$G$"$k(B. $BFsHVL\$N<0$G(B 
$q_b/a_b = 1 - e_b \ll 1$$B$+$D(B$M_p \approx M_s$$B$H2>Dj$7$?(B. 
$B$3$l$OBP>]$H$J$k$[$H$s$I$NO"@1$NG/Np$h$j>.$5$$(B.

$BBh;0$N>r7o$OH<@1$,OG@1$N6aE@:P:9(B(apsidal precession)$B$X$N<gMW$JMWAG$G$"$k(B
$B$3$H$G$"$k(B. 
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$B$7$^$&(B. $B:P:9$N;~4V(B$^5$$B$OD9$$$+$i$3$l$O87$7$$MW@A$G$"$k(B. 
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$B<!$K5s$2$k$b$N$,$"$k$H$-$G$"$k(B. (1) $B2>A[E*$JBhFs$NOG@1$^$?$O(B
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$a_X \gg a$$B$K$*$$$F$*$h$=(B
$1 \times 10^6 yr (P_{orb}/1 yr)(a_X/10 AU)^3 (1 AU/a)^3$ $(M_{Jupiter}/M_X)$
$B$G$"$k(B. $B$3$l$,(B(3)$B<0$N<~4|$rD6$($k$J$i(B, 16 Cyg B$B$NOG@1$"$k$$$O1_HW$N@-<A$K(B
$B6/$$@)8B$r2]$;$k(B. $B$?$H$($P(B, $BLZ@1%/%i%9$NJL$NOG@1$,(B
$\leq 30 AU (a_b/1000 AU)^{1/2} (q_b/100 AU)^{1/2}$$B$K$OB8:_$7F@$J$$(B. 
(2) $B0lHLAjBPO@E*8z2L(B. $B%K%e!<%H%s$N1?F0J}Dx<0$X$NAjBPO@E*Jd@5$K$h$k6aE@$N(B
$B:P:9<~4|$O<!$GM?$($i$l$k(B.$^9$ 
$$
 P_{gr} = \displaystyle 
\frac{2 \pi c^2(1-e^2) a^{5/2}}{3(GM_p)^{3/2}} = 
\displaystyle 3.36 \times 10^7 yr (1-e^2)
\left( \frac{P_{orb}}{1 yr} \right)
\left( \frac{a}{1 AU} \right)
\left( \frac{M_{\odot}}{M_p} \right) .    \eqno(4)
$$
$B$3$3$G(B$c$$B$O8wB.(B, $M_{\odot}$$B$OB@M[<ANL$G$"$k(B. 16 Cyg Bb$B$N>l9g$K$3$l$,(B
(3)$B<0$N<~4|$rD6$($k$J$i(B
$(a_b/1000 AU) (q_b/100 AU) \leq 3$$B$G$J$1$l$P$J$i$J$$(B. 

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($a=0.0462 AU, e \approx 0$)$B$K$bB8:_$9$k(B. $B$J$<$3$l$i$N50F;$O1_$K6a$$$N$+!)(B
$B<g@1$+$i$ND,<.NO$K$h$C$F50F;$,1_$K6a$E$/;~4V$O(B, $B$*$=$i$/(B, $\tau$ Boo$B$G$O(B
$B<wL?$h$jC;$/(B, 55 Cnc$B$G$OD9$$(B($B;29MJ88%(B10). $B$@$+$i>/$J$/$H$b(B55 Cnc$B$N>l9g$O(B 
$B@bL@$,I,MW$G$"$k(B. $B$?$^$?$^(B55 Cnc$B$ON%?4N($,>.$5$$0LAj$K$$$k$H$$$&$@$1(B
$B$J$N$+$b$7$l$J$$$,(B, $B$b$C$H$^$7$J@bL@$,$"$k(B. 55 Cnc$B$NO"@1$O(B$1150 AU$$BN%$l$F(B
$B$$$k(B. $a_b = 1150 AU, q_b = 337 AU$($BM}O@E*$JCf1{CM$N4|BTCM(B
$e_b = 1/\sqrt{2}$), $B$*$h$S(B$M_p = M_S = M_{\odot}$$B$r2>Dj$9$l$P(B, 
$P_{precess} \approx 2 \times 10^{10} yr$$B$*$h$S(B
$P_{gr} = 6.7 \times 10^4 yr$$B$rF@$k(B. 
$B$@$+$iAjBPO@E*:P:9$OH<@1$K$h$C$FM65/$5$l$k$b$N$h$j$b$:$C$HB.$/(B, 
$B$?$H$(JL$NOG@1$,$J$/$H$b9b$$N%?4N($OH/E8$7F@$J$$(B. 

$BN%?4N(?6F0$K$h$C$F$b$&$R$H$D8@$($k$3$H$O(B, $B2?$i$+$NJL$N%W%m%;%9$,$J$$8B$j(B,
$BO"@17O$NOG@1$N(B$\sim 1 \% $($B=i4|79<P3Q$,(B$90^{\circ}$$B$N$b$N(B)$B$O(B, 
$B$$$:$l$O<g@1$K>WFM$7$F$7$^$&$3$H$G$"$k(B.
$B=i4|$KMpJbE*$K79$$$?50F;LL>e$r$[$\1_50F;$GOG@1$,F0$1$P(B, $B:G>.6a@1E@$,(B
$<q$$B$G$"$k3d9g$O(B, $B$o$l$o$l$N2r@OE*%b%G%k$G$O(B
$f = (6q/5a)^{1/2} = 0.075 (q/R_{\odot})^{1/2} (1 AU/a)^{1/2}$$B$G$"$k(B. 
$B$?$@$7(B$R_{\odot}$$B$OB@M[H>7B$G$"$k(B. $B$7$+$7$3$N7k2L$OAjBPO@E*8z2L$rL5;k$7$F(B
$B$*$j(B, $B$3$N8z2L$ON%?4N($H$H$b$K5^B.$KA}Bg$9$k$+$i(B,  
$BB?$/$N7O$K$*$$$F1JG/?J2=$rDL$7$FC#@.$5$l$kN%?4N($K>e8B$r@_$1$k(B. 

%\vspace{0.5cm}
\newpage
\noindent
{\bf $B;29MJ88%(B}

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