{VERSION 5 0 "IBM INTEL NT" "5.0" } {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 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" 18 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 281 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 } {PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 8 8 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 18 "Area of an Eclipse" }} {PARA 19 "" 0 "" {TEXT -1 60 "by Jason Schattman, Waterloo Maple, jsch attman@maplesoft.com" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 189 "During a n eclipse of one planet by another, how does the visible area of the e clipsed planet change as viewed from earth? We seek the answer in ter ms of the apparent radii of the planets, " }{TEXT 256 1 "r" }{TEXT -1 5 " and " }{TEXT 257 1 "R" }{TEXT -1 28 ", and the apparent distance \+ " }{TEXT 258 1 "d" }{TEXT -1 21 " between the centers." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "restar t; r := 1: R := 2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 266 "for \+ i from 1 to 50 do\n c1 := plottools[disk]([0,0], R, color=green):\n \+ c2 := plottools[disk]([-.9*(R-r)-2*r*i/50,0], r, color=red):\n frame[ i] := plots[display](c2,c1):\nend do:\nplots[display](seq(frame[i],i=1 ..50), insequence=true, scaling=constrained, axes=none);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "In terms of the animation above, we seek \+ the area of the visible green region." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 162 "Maple assumes by default that all variab les are complex numbers with no relations between them, so we must tel l Maple our assumptions explicitly. In particular, " }{XPPEDIT 261 0 "r >0" "6#2\"\"!%\"rG" }{TEXT -1 2 ", " }{XPPEDIT 262 0 "R>=r" "6#1%\" rG%\"RG" }{TEXT -1 6 ", and " }{TEXT 259 1 "d" }{TEXT -1 7 " is in " } {XPPEDIT 260 0 "[R-r, R+r]" "6#7$,&%\"RG\"\"\"%\"rG!\"\",&F%F&F'F&" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "assume( r > 0, R >= r, d <= R+r, d >= R-r ): interface(showassumed=0);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 97 "Let's rotate the situation 90 degrees so that we can t reat the intersecting arcs as functions of " }{TEXT 263 1 "x" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 196 "a1 := plottools [arc]([0,0], 1, 0..Pi, color=red, thickness=2): \na2 := plottools[arc] ([0,2], 1.8, 5*Pi/4..7*Pi/4, color=green, thickness=2):\nplots[display ](a1,a2, scaling=constrained, axes=normal);" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 52 "We can express the lower arc of the green circle as " } {XPPEDIT 256 0 "y1 = d-sqrt(R^2-x^2);" "6#/%#y1G,&%\"dG\"\"\"-%%sqrtG6 #,&*$%\"RG\"\"#F'*$%\"xGF.!\"\"F1" }{TEXT -1 40 " and the upper arc of the red circle as " }{XPPEDIT 257 0 "y2 = sqrt(r^2-x^2);" "6#/%#y2G-% %sqrtG6#,&*$%\"rG\"\"#\"\"\"*$%\"xGF+!\"\"" }{TEXT -1 77 ". Our strat egy is to compute the area of the eclipsed region by integrating " } {TEXT 278 7 "y2 - y1" }{TEXT -1 65 " between their intersection points . Subtracting this result from " }{XPPEDIT 256 0 "Pi*R^2" "6#*&%#PiG\" \"\"*$%\"RG\"\"#F%" }{TEXT -1 55 " will give us the visible area of th e eclipsed planet. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "y1 : = d - sqrt(R^2-x^2);\ny2 := sqrt(r^2-x^2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "The difference between th e curves." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "EclipseWidth : = y2-y1;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Now we let Maple compute the intersection points of the c ircles by solving " }{TEXT 276 7 "y1 = y2" }{TEXT -1 5 " for " }{TEXT 277 1 "x" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "x1,x2 := solve( y1=y2, x );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "Integrating the difference between the curves." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 48 "Int( EclipseWidth, x ) = int( EclipseWidth, x \+ );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "For Maple-convenience, we make an explicit function " }{TEXT 267 4 "g(x)" }{TEXT -1 41 " out of the integral above using Maple's " }{TEXT 266 7 "unapply" }{TEXT -1 10 " command.." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 24 "g := unapply(rhs(%), x):" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 164 "By symmetry about the y-axis, the area is double the definite integral from 0 to the ri ght-most intersection point. Therefore, the formula for the visible a rea is " }{XPPEDIT 265 0 "Pi*R^2 - 2*(g(x1)-g(0))" "6#,&*&%#PiG\"\"\"* $%\"RG\"\"#F&F&*&F)F&,&-%\"gG6#%#x1GF&-F-6#\"\"!!\"\"F&F3" }{TEXT -1 3 ". " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Pi*R^2 - 2*(g(x1) -g(0));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "So that we can easily plot this formula, we again make th e area formula an explicit function of " }{TEXT 268 4 "R, r" }{TEXT -1 5 " and " }{TEXT 269 1 "d" }{TEXT -1 11 " using the " }{TEXT 264 7 "unapply" }{TEXT -1 8 " command" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "VisibleArea := unapply( %, R, r, d ):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "When " }{TEXT 279 5 "R = r" }{TEXT -1 65 ", the visible area of the green planet is \+ strictly increasing in " }{TEXT 280 1 "d" }{TEXT -1 4 ". " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "plot( VisibleArea(1, 1, d), \+ d=0..2, thickness=2, color=blue);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "But as soon as " }{TEXT 270 5 "R > r" }{TEXT -1 57 ", the behavior changes radically! Notice the area first " }{TEXT 271 9 "decreases" }{TEXT -1 84 " as the eclipsing planet begins to move away. Here's a \+ comparison of the two cases " }{TEXT 273 9 "R = r = 1" }{TEXT -1 15 " \+ (in blue) and " }{TEXT 272 16 "R = 1, r = .95 " }{TEXT -1 88 "(in bro wn). Study the animation above again to get an intuition for why this happens. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "plot( [Visib leArea(1, 1, d), VisibleArea(1, .95, d)], d=0..2, thickness=3, color=[ blue,orange]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "We can also und erstand this in terms of the area formula for the cases " }{TEXT 281 5 "R = r" }{TEXT -1 5 " vs. " }{TEXT 282 5 "R > r" }{TEXT -1 95 ". No te that the formula in the first case is much simpler because many ter ms have dropped out." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Vis ibleArea( R, R, d );" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "VisibleArea( R, r , d );" }}}{EXCHG }{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "The cases " } {TEXT 274 22 "r = 1, 1.25, 1.5, 1.75" }{TEXT -1 5 " and " }{TEXT 275 1 "2" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "plo t( [seq(VisibleArea(1+i/4, 1, d), i=0..4)], d=0..3, thickness=2);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }