1 Formulas. (%i1) kill(all); (%o0) done. (%i1) numer:false; (%o1) false. (%i2) drdt1: epsilonxl/(alphamu)sin(x*theta); (%o2) ε x sin( θ x)

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Download "1 Formulas. (%i1) kill(all); (%o0) done. (%i1) numer:false; (%o1) false. (%i2) drdt1: epsilon*x*l/(alpha*mu)*sin(x*theta); (%o2) ε x sin( θ x)"

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1 (5).wxm 1 / 1 1 Formulas (%i1) kill(all); (%o0) done (%i1) numer:false; (%o1) false (%i) drdt1: epsilon*x*l/(alpha*mu)*sin(x*theta); (%o) ε x sin( θ x) L αμ (%i3) drdt: x*l/(alpha*mu*r)*(epsilon^*r^-(alpha-r)^)^(1/); (%o3) ε r -( α -r) xl αμr (%i4) dadr1: alpha/(*epsilon*sin(x*theta)); α (%o4) ε sin( θ x) (%i5) dadr: alpha/(*x)*r/(epsilon^*r^-(alpha-r)^)^(1/); α r (%o5) ε r -( α -r) x (%i6) f1: (alpha/(1+epsilon*cos(x*theta)))^; (%o6) α ( ε cos( θ x ) +1) (%i7) assume(epsilon^-1<0); (%o7) [ ε <1] (%i8) A: x/alpha*integrate(f1, theta); (%o8) α atan ( ε - ) sin( θ x) 1 - ε ( cos( θ x ) +1) ε sin( θ x) ε ( ε - 1) ( cos( θ x ) +1) ( ε 3 - ε - ε + 1) sin( θ x) - ( cos( θ x ) +1) ε 3 - ε + ε + 1 (%i9) f: r/(epsilon^*r^-(alpha-r)^)^(1/); r (%o9) ε r -( α -r)

2 (5).wxm / 1 (%i10) assume(epsilon>0, alpha>0); (%o10) [ ε >0,α >0] (%i11) Ar: alpha/(*x)*integrate(f,r); ( ε α asin - 1) r+ α 4 α ( ε - 1 ) + 4 α α + ( ε - 1 ) r + α r-α 1 - ε ( ε - 1) ε - 1 (%o11) x Plots.1 dr/dt (r) (%i1) P11: ev(drdt, [L=1, mu=1, alpha=1, epsilon=0.3, x=0.7]); P1: ev(drdt, [L=1, mu=1, alpha=1, epsilon=0.3, x=1]); P13: ev(drdt, [L=1, mu=1, alpha=1, epsilon=0.3, x=1.3]); (%o1) r -( 1 -r) r (%o13) 0.09 r -( 1 -r) r (%o14) r -( 1 -r) r (%i15) wxplotd([p11,p1,p13], [r,0.7,1.5], [xlabel, "r"], [ylabel, "dr/dt"], [legend, "x=0.7", "x=1.0", "x=1.3"]); (%t15) (%o15)

3 (5).wxm 3 / 1. dr/dt (theta) polar plot (%i16) P11: ev(drdt1, [L=1, mu=1, alpha=1, epsilon=0.3, x=0.7]); P1: ev(drdt1, [L=1, mu=1, alpha=1, epsilon=0.3, x=1]); P13: ev(drdt1, [L=1, mu=1, alpha=1, epsilon=0.3, x=1.3]); (%o16) 0.1 sin( ) 0.7 θ (%o17) 0.3 sin( θ) (%o18) 0.39 sin( 1.3 θ) (%i19) wxplotd([p11,p1,p13], [theta,0,1.5*%pi], [xlabel, "theta"], [ylabel, "dr/dt"], [legend, "x=0.7", "x=1.0", "x=1.3"]); (%t19) (%o19)

4 (5).wxm 4 / 1 (%i0) wxdrawd(user_preamble = "set grid polar; set size square;", nticks = 1000, line_width =, xrange = [-.5,.5], yrange = [-.5,.5], /*title = "Precessing Ellipse, orbits",*/ = "x=0.7", = blue, polar(p11, theta, 0.1, 4*%pi), = "x=1.0", = red, polar(p1, theta, 0.1, 4*%pi), = "x=1.3", = green, polar(p13, theta, 0.1, 4*%pi) ); (%t0) (%o0) [ grd( polar, polar, polar )].3 da/dr(theta) polar plot (%i1) P11: ev(dadr1, [alpha=1, epsilon=0.3, x=0.7]); P1: ev(dadr1, [alpha=1, epsilon=0.3, x=1]); P13: ev(dadr1, [alpha=1, epsilon=0.3, x=1.3]); (%o1) sin( 0.7 θ) (%o) sin( θ) (%o3) sin( 1.3 θ)

5 (5).wxm 5 / 1 (%i4) wxplotd([p11,p1,p13], [theta,0,1.5*%pi], [y,0,10], [xlabel, "theta"], [ylabel, "da/dr"], [legend, "x=0.7", "x=1.0", "x=1.3"]); plotd: some values were clipped. plotd: some values were clipped. plotd: some values were clipped. (%t4) (%o4)

6 (5).wxm 6 / 1 (%i5) wxdrawd(user_preamble = "set grid polar; set size square;", nticks = 1000, line_width =, xrange = [-5.5,5.5], yrange = [-5.5,5.5], /*title = "Precessing Ellipse, orbits",*/ = "x=0.7", = blue, polar(p11, theta, 0.1, *%pi), = "x=1.0", = red, polar(p1, theta, 0.1, *%pi), = "x=1.3", = green, polar(p13, theta, 0.1, *%pi) ); (%t5) (%o5) [ grd( polar, polar, polar )].4 da/dr(r) (%i6) P1: ev(dadr, [alpha=1, epsilon=0.3, x=0.7]); P: ev(dadr, [alpha=1, epsilon=0.3, x=1]); P3: ev(dadr, [alpha=1, epsilon=0.3, x=1.3]); (%o6) r 0.09 r -( 1 -r) r (%o7) 0.09 r -( 1 -r) (%o8) r 0.09 r -( 1 -r)

7 (5).wxm 7 / 1 (%i9) wxplotd([p1,p,p3], [r,0.7,1.7], [y,0,10], [xlabel, "r"], [ylabel, "da/dr"], [legend, "x=0.7", "x=1.0", "x=1.3"]); plotd: some values were clipped. plotd: some values were clipped. plotd: some values were clipped. (%t9) (%o9).5 A(theta) polar plot

8 (5).wxm 8 / 1 (%i30) P31: ev(a, [alpha=1, epsilon=0.3, x=0.7]); P3: ev(a, [alpha=1, epsilon=0.3, x=1]); P33: ev(a, [alpha=1, epsilon=0.3, x=1.3]); (%o30) ( atan sin( 0.7 θ ) cos( 0.7 θ ) sin( 0.7 θ) ( cos( 0.7 θ ) + 1) ) sin( 0.7 θ) ( cos( 0.7 θ ) + 1) (%o31) ( atan sin( θ ) cos( θ ) sin( θ) ( cos( θ ) + 1) ) sin( θ) ( cos( θ ) + 1) (%o3) ( atan sin( 1.3 θ ) cos( 1.3 θ ) sin( 1.3 θ) ( cos( 1.3 θ ) + 1) ) sin( 1.3 θ) ( cos( 1.3 θ ) + 1) (%i33) wxplotd([p31,p3,p33], [theta,0,1.5*%pi], [xlabel, "theta"], [ylabel, "A(theta)"], [legend, "x=0.7", "x=1.0", "x=1.3"]); (%t33) (%o33)

9 (5).wxm 9 / 1 (%i34) wxdrawd(user_preamble = "set grid polar; set size square;", nticks = 1000, line_width =, xrange = [-5.5,5.5], yrange = [-5.5,5.5], /*title = "Precessing Ellipse, orbits",*/ = "x=0.7", = blue, polar(p31, theta, 0.1, *%pi), = "x=1.0", = red, polar(p3, theta, 0.1, *%pi), = "x=1.3", = green, polar(p33, theta, 0.1, *%pi) ); (%t34) (%o34) [ grd( polar, polar, polar )].6 A(r) (%i35) P31: ev(ar, [alpha=1, epsilon=0.3, x=0.7]); P3: ev(ar, [alpha=1, epsilon=0.3, x=1]); P33: ev(ar, [alpha=1, epsilon=0.3, x=1.3]); (%o35) ( r + r asin( ( r ) ) (%o36) ( r + r asin( ( r ) )/ (%o37) ( r + r asin( ( r ) )

10 (5).wxm 10 / 1 (%i38) wxplotd([p31,p3,p33], [r,0.7,1.7], [xlabel, "r"], [ylabel, "A(r)"], [legend, "x=0.7", "x=1.0", "x=1.3"]); (%t38) (%o38).7 A(r,x) for r=rmin, r=alpha, r=rmax (%i39) P31: ev(ar, [alpha=1, epsilon=0.3, r=1/(1+0.3)+0.001]); P3: ev(ar, [alpha=1, epsilon=0.3, r=1]); P33: ev(ar, [alpha=1, epsilon=0.3, r=1/(1-0.3)-0.001]); (%o39) x (%o40) x (%o41) x

11 (5).wxm 11 / 1 (%i4) wxplotd([p31,p3,p33], [x,0,3.5], [y,-10,10], [xlabel, "x"], [ylabel, "A(x)"], [legend, "r=r_min", "r=alpha", "r=r_max"]); plotd: some values were clipped. plotd: some values were clipped. plotd: some values were clipped. (%t4) (%o4).8 Atot(x) (%i43) numer:true; (%o43) true lower integration limit (%i44) P31: ev(a, [alpha=1, epsilon=0.3, x=0.7, theta=0]); P3: ev(a, [alpha=1, epsilon=0.3, x=1, theta=0]); P33: ev(a, [alpha=1, epsilon=0.3, x=1.3, theta=0]); (%o44) 0.0 (%o45) 0.0 (%o46) 0.0 upper integation limit (%i47) P31: ev(a, [alpha=1, epsilon=0.3, x=0.7, theta=1*%pi]); P3: ev(a, [alpha=1, epsilon=0.3, x=1, theta=1*%pi ]); P33: ev(a, [alpha=1, epsilon=0.3, x=1.3, theta=1*%pi]); (%o47) (%o48) (%o49)

12 (5).wxm 1 / 1 (%i50) Ax: * ev(a, [alpha=1, epsilon=0.3, theta=%pi]); (%o50) 4 ( atan sin( x) - cos( x ) sin( x) ( cos( x ) +1) ) sin( x) ( cos( x ) +1) (%i51) wxplotd([ax], [x,0,3.5], [xlabel, "x"], [ylabel, "A_tot(x)"]); (%t51) (%o51)

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