{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 } {PSTYLE "Heading 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 }{PSTYLE "Author" 0 19 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 8 8 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 4" -1 257 1 {CSTYLE "" -1 -1 "Times" 0 20 0 0 0 0 2 1 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 0 0 }{PSTYLE "R3 Font 5" -1 258 1 {CSTYLE "" -1 -1 "Times" 0 17 0 0 0 0 2 1 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 0 0 }{PSTYLE "Share Details" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 3 80 0 0 0 0 0 0 }{PSTYLE " " 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 266 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 267 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 268 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 69 "Perihelion precession rate of Mercury according to General Relativity " }}{PARA 19 "" 0 "" {TEXT -1 30 "by Tony Scott and Blair Madore" }} {PARA 259 "" 0 "" {TEXT 256 9 "Abstract:" }{TEXT -1 125 " Worksheet th at shows a calculation for the perihelion precession rate of the plane t Mercury according to general relativity." }}{PARA 259 "" 0 "" {TEXT 259 28 "Application Areas/Subjects: " }{TEXT 257 0 "" }{TEXT -1 61 " S cience, Astronomy & Astrophysics, Physics, Applied Examples" }}{PARA 259 "" 0 "" {TEXT 258 9 "Keywords:" }{TEXT -1 74 " General relativity , relativity, planet, Mercury, precession, perihelion," }}{PARA 259 " " 0 "" {TEXT -1 0 "" }{TEXT 256 9 "See Also:" }{TEXT -1 32 " MapleTec h Special issue 1994. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 " Introduction" }}{PARA 0 "" 0 "" {TEXT -1 327 "The planetary motions are essentially describe d by ellipses although they may be slightly perturbed by the motion of the other \"heavenly bodies\". First, consider only the interaction b etween the sun's gravitational field and any given planet (or asteroid ): we use a spherical coordinate system about the sun with coordinates : " }{XPPEDIT 18 0 " [r, theta, phi]" "6#7%%\"rG%&thetaG%$phiG" }} {PARA 0 "" 0 "" {TEXT -1 140 "We examine only the motion of r vs. th eta and for the non-relativistic case, the equation of motion (for th e Kepler orbits) is given by: " }}{PARA 260 "" 0 "" {TEXT -1 0 "" } {XPPEDIT 18 0 "diff(u(theta),`$`(theta,2))+u(theta) = A" "6#/,&-%%diff G6$-%\"uG6#%&thetaG-%\"$G6$F+\"\"#\"\"\"-F)6#F+F0%\"AG" }}{PARA 0 "" 0 "" {TEXT -1 84 "Where A is an expression in e , a , m , and mu - the reduced mass of the planet." }}{PARA 0 "" 0 "" {TEXT -1 108 "We t ake this opportunity to define precisely the meaning of these symbols \+ and others used in this worksheet:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 27 "Symbol , Units, and Meaning" }}{PARA 0 " " 0 "" {TEXT -1 87 "-------------------------------------------------- -------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 94 " Gravitational constant. G=6.672*10^(-11) \+ Newton-meters^2/Kg^2 " }}{PARA 0 "" 0 "" {TEXT -1 75 "Mass of the S un. M = 1.99*10^30 Kg " }} {PARA 0 "" 0 "" {TEXT -1 62 "orbit Eccentricity of Mercury \+ e = 0.2056 " }}{PARA 0 "" 0 "" {TEXT -1 75 "semi-major axis \+ of ellipse traced by Mercury's orbit a = 0.3871 A.U " }}{PARA 0 "" 0 "" {TEXT -1 75 "Astronomical Unit for distance. \+ A.U.=1.495*10^11 meters " }}{PARA 0 "" 0 "" {TEXT -1 92 "speed of \+ light. c =2.997925*10^8 mete rs/second " }}{PARA 0 "" 0 "" {TEXT -1 72 "Period of Mercury's orbi t. T = 0.2408 years " }}{PARA 0 "" 0 "" {TEXT -1 87 "--------------------------------------------------------- ------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 16 "Now we see A is " }}{PARA 262 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "A = m/mu/a/(1-e^2)" "6#/%\"AG**%\"mG\"\" \"%#muG!\"\"%\"aGF),&F'F'*$%\"eG\"\"#F)F)" }}{PARA 0 "" 0 "" {TEXT -1 34 "The reduced mass mu is defined as " }}{PARA 261 "" 0 "" {XPPEDIT 18 0 "m/(1+m/M)" "6#*&%\"mG\"\"\",&F%F%*&F$F%%\"MG!\"\"F%F)" }}{PARA 0 "" 0 "" {TEXT -1 104 "Since the mass of the sun is much larger than \+ the planets m/mu is approximately 1. So A is approximately" }}{PARA 263 "" 0 "" {XPPEDIT 18 0 "1/(a*(1-e^2))" "6#*&\"\"\"F$*&%\"aGF$,&F$F$ *$%\"eG\"\"#!\"\"F$F+" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "The solution to this equation is well known." }}{PARA 264 "" 0 "" {XPPEDIT 18 0 "u(theta) = A*(1+C*cos(theta))" "6#/-%\"uG6# %&thetaG*&%\"AG\"\"\",&F*F**&%\"CGF*-%$cosG6#F'F*F*F*" }}{PARA 0 "" 0 "" {TEXT -1 179 "and this constant called e defines the eccentricity \+ of the orbit. The above is determined when at theta = 0, u is at a max imum and r is a minimum. This corresponds to perihelion." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 164 "This is an oscill ary solution with natural angular frequency of 1 and every 2 Pi rotat ion of the planet around the Sun returns u (and r) to its perihelion value." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 299 "However if we include the effects of other bodies, this causes pe rturbations in such a way that u(theta) does NOT quite return to it's \+ initial value and thus the perihelion is displaced and the major axis \+ of the ellipse starts moving at each revolution. In other words, the o rbits starts precessing." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 329 "These effects are very small. The perihelion of M ercury, which shows the largest effect, was observed to move only abou t 574 sec of arc per century. Detailed calculations of the influence o f the other planets on the motion of Mercury predict that the precessi on rate of the perihelion should be about 531 sec of arc per century. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "This left a discrepancy of 43 sec which could not be accounted for! (using standard Newtonian mechanics)." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 65 " What happens when we cons ider the effects of General Relativity?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "Starting from the Schwarzchil d metric one can show that our equation of motion is modified to:" }} {PARA 265 "" 0 "" {XPPEDIT 18 0 "diff(diff(u(theta),theta),theta)+u(th eta) = A+epsilon*u(theta)^2/A" "6#/,&-%%diffG6$-F&6$-%\"uG6#%&thetaGF- F-\"\"\"-F+6#F-F.,&%\"AGF.*(%(epsilonGF.*$-F+6#F-\"\"#F.F2!\"\"F." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "Here epsi lon (a small amount) is given in terms of G , M , and c as defined ab ove." }}{PARA 266 "" 0 "" {XPPEDIT 18 0 "epsilon = 3*G*M/a/c^2/(1-e^2) " "6#/%(epsilonG*.\"\"$\"\"\"%\"GGF'%\"MGF'%\"aG!\"\"*$%\"cG\"\"#F+,&F 'F'*$%\"eGF.F+F+" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 312 "Our equation has an additional inhomogeneous term which \+ is proportional to u^2 and consequently the DE is non linear. In the r est of this Maple session we will use a method of Kryloff and Bogoliub off to find a series solution and thus calculate the precession rate \+ of Mercury as predicted by General Relativity." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "First we use the alias co mmand to ensure the output is printed correctly." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "alias(w[0]=w0, w[1]=w1,u[0]=u0,u[1]=u1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "First we set up the basic differential equation governing the planetary orbits in General Relativity" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 75 "eq:= diff(u(theta),theta$2) + (w[0]^2)*u(theta ) = A + epsilon*u(theta)^2/A;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 157 "Here u = 1/r where r is the distance from the planet to the Sun, epsi lon is defined as above, and A is a non-relativistic inhomogeneous ter m (i.e. constant)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "Let's solve for w[0]=1 and epsilon=0 (a non-relativistic \+ limit)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "dsolve(subs(epsilon=0,w[0]=1,%), u(theta));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 170 "So the zero-order solution is that of a simple harmonic oscillator with a natural angular frequency w[0]=1 plus a co nstant. The MAPLE constants _C1 and _C2 are arbitrary." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 220 "Thus for non-relati vistic Keplerian motion, r(theta) returns exactly to it's original pos ition every 2 Pi radians. However the non-linear term will cause a fre quency shift from w[0]=1 and consequently cause a precession." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "We now ge t a series solution in powers of \"epsilon\" using the method of Krylo ff and Bogoliuboff:" }}{PARA 0 "" 0 "" {TEXT -1 36 "Expand u(theta) an d frequency w[0] :" }}{PARA 268 "" 0 "" {TEXT -1 8 " " } {XPPEDIT 18 0 " u(theta) = u[0](theta) + epsilon u[1](theta)" "6#/-%\" uG6#%&thetaG,&-&F%6#\"\"!6#F'\"\"\"*&%(epsilonGF.-&F%6#F.6#F'F.F." } {TEXT -1 6 " + ..." }}{PARA 267 "" 0 "" {TEXT -1 18 " \+ " }{XPPEDIT 18 0 "w[0]^2 = w^2 + epsilon w[1] " "6#/*$&%\"wG6#\"\"! \"\"#,&*$F&F)\"\"\"*&%(epsilonGF,&F&6#F,F,F," }{TEXT -1 5 "+ ..." }} {PARA 0 "" 0 "" {TEXT -1 27 "We achieve this as follows:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "eval( subs( u(theta)=u[0](theta)+epsilon*u[1] (theta), eq ) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "subs(w[ 0]^2=w^2+epsilon*w[1]+epsilon^2*w[2],%):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "We obtain a series expansion in epsilon on both sides" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "eqs :=map(series,%,epsilon, 2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Co llect the zero-order equation in eq[0]," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "eq[0]:=coeff(lhs(eqs),epsilon, 0)= coeff(rhs(eqs),epsilon,0);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "and the first order equation in eq[1]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "eq[1]:=coeff(lhs(eqs),epsilon, 1)= coeff(rhs(eqs),epsilon,1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Now solve the zero-order equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "dsolve(eq[0],u[0](theta));" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 222 "We obtain essentially the same so lution as before. We will assign this solution to u[0] and simplify it by insisting that the angle theta be measured from the position of th e perihelion (i.e. u[0] is a maximum at theta=0)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "assign(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "We find the derivative w.r.t. theta of u[0](theta) at theta =0." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "eva l(subs(theta=0,diff(u[0](theta),theta)));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "Thus to insure u[0](theta) is a maximum at theta=0 we set the constant _C2 to zero." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "_C2:=0:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 135 "Next we solve the first-order equation after simplifying the inho mogeneous term. We use combine with it's trigonometry option to do so. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "eq[1]:=combine( eval(eq[1]) , t rig );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "dsolve( eq[1], u[ 1](theta) );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 150 "Maple has given \+ us a particular solution plus a complementary solution with arbitrary \+ constants _C3 and _C4, which we don't need, and thus throw away:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "sub s (_C3=0, _C4=0, %):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Now we co llect in theta." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "collect( combine( % , trig) , theta );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "We note that Maple has not made the simpl ification" }}{PARA 0 "" 0 "" {TEXT -1 134 "This is of course correct, \+ since arctan(tan(x))=x is not true in general. In this case it is OK \+ to make the simplification, so we do." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "arctan(sin(w*theta/2)/cos(w*theta/2)):=w*theta/2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "collect(%%,theta);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 327 "Now we notice terms like cos(theta) and \+ cos(2*theta) and these represent a small periodic disturbance of the \+ normal Keplerian motion. They will not contribute on average. However , the term proportional to \"theta\" is secular and therefore observab le effects arise on an sufficiently long period of time. So we grab t his term." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "op(1,rhs(%));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "We solve for w[1] such that this term is zero." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "w[1] := solve( % , \+ w[1] );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "To first-order in epsi lon, we can isolate a value for w." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "### WARNING: persistent store makes one-argument readlib obsolete\nreadlib(isolate):" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 43 "isolate( w[0]^2 = w^2 + w[1]*epsilon , w ); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "And w[0]=1 and to first-order in epsilon the effective frequency w is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "simplify( subs( w[0]=1 , w = series(rhs(%),epsilon,2) ) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "assign(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "Thus cos(w theta) has a maximum when" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 17 "w*theta = 2*Pi*n;" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 55 "Where n is a non-negative integer. Let's isolate theta. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "isolate(%,theta);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "To first or der in \"epsilon\", this is just" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "lhs(%) = series(rhs(%),epsilon,2); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 140 "Thus the successive periheli a do not appear at intervals of 2 Pi but, rather of 2 Pi + delta which is given by the second term on the right." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "delta:=subs(n=1,coeff(rhs (%),epsilon,1))*epsilon;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Subst ituting the value of epsilon in the above" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "delta:=subs(epsilon = 3*G *M/(a*c^2*(1-e^2)),%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "Finally , we get the well-known formula for the perihelion precession rate. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 274 "We see therefore that the effect is enhanced if the semimajor axis a is smal l and the eccentricity \"e\" is large. Mercury, which is the planet ne arest the Sun and which has the most eccentric orbit of any planet (a fter Pluto), provides the most sensitive test of the theory." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Next we plug in values for the gravitational constant G and the sun's mass M." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "sub s(G=6.672*(10^(-11)),M=1.99*(10^(30)),%):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "Then, plug in quantities for the orbit parameters of Merc ury" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "delta[Mercury]:=subs(e=.2056, a=.3871*1.495*(10^11),c=2.997925*(10 ^8),%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 229 "Now this is a very sm all arc, so we consider the change of arc over a century. In this cas e, Mercury will have made (100/T) revolutions and the result in radian s has to be converted into seconds of arc. T is the period of orbit." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "T :=0.2408:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "evalf(3600*(36 0/(2*Pi))*(100/T)*delta[Mercury],4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 " Summary & Refere nces" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 207 " Thus Mercury's precessional rate in seconds of arc per century due to \+ relativistic effects is 43.03. The observed value is 43.11 so that the prediction of General Relativity is confirmed in striking fashion." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 10 " References" }}{PARA 0 "" 0 "" {TEXT -1 116 "[1] J.B. Marion, Classical Dynamics of Particles and Systems, Academic Press, 2nd ed., pp. 167-1 70, 267-270 (1970)." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "14" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }