#OK to post homework
#GLORIA LIU, Apr 9 2024, Assignment 22

read `C22.txt`:

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#1. Read and understand the wikipedia article on solar eclipse
#Done

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#2. Read and understand the wikipedia article on Tidal force and convince youself that the tidal strenth of the heavenly body on the earth is 
#proportional to the mass divided by the cube of the distance to the earth from that object. Explain how this fact, combined with the fact 
#that the apparent sizes of the moon and the sun to an observer on Earth is about the same (making at total #eclipse possible, but just 
#barely) combined with the known fact that the lunar tide is much stronger #than the solar tide, proves that the density (mass per volume) of
#the moon is significantly greater #than that of the sun.

#Let vm, vs be the volume of the moon and sun respectively, and rm and rs be the radius of the moon and sun respectively
#Let mm, ms be the mass of the moon and sun, dm and ds be the distance from the earth to the moon and sun respectively

#We want to find the relation between mm/vm and ms/vs, and show that mm/vm is significantly greater than ms/vs. 
#The formula for the volume of a sphere is (4/3)*Pi*r^3 where r is the radius of the sphere. In our case, we will be looking at the volume
#of two spheres, so we could just look at the radius cubed, as the constants will not affect the ratio of mm/vm to ms/vs.
#From the possibility of the eclipse, we have that roughly, rs/ds = rm/dm. So ds is roughly equal to (rs*dm)/rm.
#From the strength of the tidal forces, we have mm/(dm^3) is significantly greater than ms/(ds^3).
#Substituting in ds=(rs*dm)/rm in the tidal forces equation we obtain mm/(dm^3) >> ms*(rm/(rs*dm))^3, 
#Equivalently mm/(dm^3) >> (ms*rm^3)/(rs^3*dm^3), which is equivalent to mm/(rm^3) >> ms/(rs^3), which implies that mm/vm >> ms/vs.
#So the density of the moon is significantly greater than that of the sun.


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#3. Start working on your chosen class final project and report the progress so far

#Progress: met with Dr. Z during office hours and learned how to use the Maple package to find recurrence formula.

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#4. Complete procedure SE(SunCenter,SunRadius,MoonCenter,MoonRadius,EarthCenter,EarthRadius) (that we barely started in C22.txt , to draw a 
#diagram similar to the one in the "Geometry" section of the wikipedia article on solar eclipse

with(geometry):
#with(Student:-Precalculus):
with(plottools):
with(plots):

###Important!!! Please restart maple environment if you want to use this process more than once###

SEgl:=proc(SunCenter,SunRadius,MoonCenter,MoonRadius,EarthCenter,EarthRadius) local P,t,Dse,Dem,simCoord,inSimCoord,exSimCoord,eTL1,eTL2,solsEMTL1,solsEMTL2,solsESTL1,solsESTL2,iTL1,iTL2,solsIMTL1,solsIMTL2,solsISTL1,solsISTL2,solsPen1,solsPen2,penRadius:
geometry[point](A,SunCenter[1],SunCenter[2]), geometry[point](B,MoonCenter[1],MoonCenter[2]), geometry[point](C,EarthCenter[1],EarthCenter[2]):
if not AreCollinear(A,B,C) then
 RETURN(FAIL):
fi:

#Plot the sun
P:=plot([SunCenter[1]+SunRadius*cos(t),SunCenter[2]+SunRadius*sin(t),t=0..2*Pi],axes=none,filled=true,color=yellow):

#Earth orbit plot
Dse:=Student[Precalculus][Distance](SunCenter,EarthCenter):
P:=P,plot([SunCenter[1] + Dse*cos(t), SunCenter[2] + Dse*sin(t), t=-Pi/6..2*Pi/3], color=blue, thickness=1):

#Moon orbit plot
Dem:=Student[Precalculus][Distance](EarthCenter,MoonCenter):
P:=P,plot([EarthCenter[1] + Dem*cos(t), EarthCenter[2] + Dem*sin(t), t=0..2*Pi], color=green, thickness=1):

#Plot the moon and earth. Unfortunately even plotting after orbits, the planets do not cover their orbit lines.
P:=P,plot([MoonCenter[1]+MoonRadius*cos(t),MoonCenter[2]+MoonRadius*sin(t),t=0..2*Pi],axes=none,filled=true,color="LightGray"):
P:=P,plot([EarthCenter[1]+EarthRadius*cos(t),EarthCenter[2]+EarthRadius*sin(t),t=0..2*Pi],axes=none,filled=true,color=blue):

#Find similitude points
_EnvHorizontalName := 'x':
_EnvVerticalName := 'y':
geometry[circle](sun, [geometry[point](A,SunCenter[1],SunCenter[2]),SunRadius]), geometry[circle](moon, [geometry[point](B,MoonCenter[1],MoonCenter[2]) ,MoonRadius]), geometry[circle](earth, [geometry[point](C,EarthCenter[1],EarthCenter[2]),EarthRadius]):
similitude(objSim,sun,moon):
simCoord:=map(coordinates,objSim):
inSimCoord:=simCoord[1]:
exSimCoord:=simCoord[2]:

#Find and graph external tangents to moon and sun
geometry[point](ISC, inSimCoord[1], inSimCoord[2]), geometry[point](ESC, exSimCoord[1], exSimCoord[2]):
TangentLine(objTL,ESC,moon,[lE1,lE2]):
eTL1:=solve(Equation(lE1),y):
eTL2:=solve(Equation(lE2),y):
solsEMTL1:=solve({Equation(lE1), (x-MoonCenter[1])^2 + (y-MoonCenter[2])^2 = (MoonRadius^2) + 0.0001}, {x,y}):
solsEMTL2:=solve({Equation(lE2), (x-MoonCenter[1])^2 + (y-MoonCenter[2])^2 = (MoonRadius^2) + 0.0001}, {x,y}):
P:=P,plot(eTL1,x=rhs(solsEMTL1[1][1])..exSimCoord[1],color=gray):
P:=P,plot(eTL2,x=rhs(solsEMTL2[1][1])..exSimCoord[1],color=gray):
solsESTL1:=solve({Equation(lE1), (x-SunCenter[1])^2 + (y-SunCenter[2])^2 = (SunRadius^2) + 0.0001}, {x,y}):
solsESTL2:=solve({Equation(lE2), (x-SunCenter[1])^2 + (y-SunCenter[2])^2 = (SunRadius^2) + 0.0001}, {x,y}):
P:=P,plot(eTL1,x=rhs(solsESTL1[1][1])..rhs(solsEMTL1[1][1]),color=yellow):
P:=P,plot(eTL2,x=rhs(solsESTL2[1][1])..rhs(solsEMTL2[1][1]),color=yellow):

#Find and graph internal tangents to moon and sun
TangentLine(objTL2,ISC,moon,[lI1,lI2]):
iTL1:=solve(Equation(lI1),y):
iTL2:=solve(Equation(lI2),y):
solsIMTL1:=solve({Equation(lI1), (x-MoonCenter[1])^2 + (y-MoonCenter[2])^2 = (MoonRadius^2) + 0.0001}, {x,y}):
solsIMTL2:=solve({Equation(lI2), (x-MoonCenter[1])^2 + (y-MoonCenter[2])^2 = (MoonRadius^2) + 0.0001}, {x,y}):
solsISTL1:=solve({Equation(lE1), (x-SunCenter[1])^2 + (y-SunCenter[2])^2 = (SunRadius^2) + 0.0001}, {x,y}):
solsISTL2:=solve({Equation(lE2), (x-SunCenter[1])^2 + (y-SunCenter[2])^2 = (SunRadius^2) + 0.0001}, {x,y}):
P:=P,plot(iTL1,x=rhs(solsISTL1[1][1])..rhs(solsIMTL1[1][1]),color=yellow):
P:=P,plot(iTL2,x=rhs(solsISTL2[1][1])..rhs(solsIMTL2[1][1]),color=yellow):
PerpendicularLine(lp,ESC,lI1):
solsPen1:=solve({Equation(lp), Equation(lI1)},{x,y}):
PerpendicularLine(lp2,ESC,lI2):
solsPen2:=solve({Equation(lp2), Equation(lI2)},{x,y}):
P:=P,plot(iTL1,x=rhs(solsIMTL1[1][1])..rhs(solsPen1[1]),color=gray):
P:=P,plot(iTL2,x=rhs(solsIMTL2[1][1])..rhs(solsPen2[1]),color=gray):

#Graph penumbra and umbra
penRadius:=geometry[distance](ESC,lI1):
P:=P,plot([exSimCoord[1]+penRadius*cos(t), exSimCoord[2]+penRadius*sin(t),t=0..2*Pi],axes=none,filled=true,color=black):
P:=P,plottools[point](exSimCoord,color=black,symbol=solidcircle):

#Graph text labels and lines pointing to penumbra/umbra
P:=P,textplot([exSimCoord[1]-Dem-.8, exSimCoord[2]+.5, "Penumbra"], font=["times","roman",bold,10],color=black,axes=none):
P:=P,textplot([exSimCoord[1]-Dem-.8, exSimCoord[2], "Umbra"], font=["times","roman",bold,10],color=black,axes=none):
P:=P,textplot([EarthCenter[1], EarthCenter[2]+Dem+.2, "Orbit of the Moon"], font=["times","roman",bold,10],color=green,axes=none):
P:=P,textplot([SunCenter[1], SunCenter[2]+Dse-.3, "Orbit of the Earth"], font=["times","roman",bold,10],color=blue,axes=none):
P:=P,textplot([SunCenter[1]-SunRadius-.5, SunCenter[2], "Sun"], font=["times","roman",bold,10],color="Gold",axes=none):
P:=P,plottools[line]([exSimCoord[1]-Dem-.4, exSimCoord[2]], exSimCoord,color=black):
P:=P,plottools[line]([exSimCoord[1]-Dem-.2, exSimCoord[2]+.5], [rhs(solsPen1[1]) + .1,rhs(solsPen1[2])],color=black):

display(P);
end:

#Try: SEgl([0, 0], 1, [4, 4], 0.2, [5, 5], 0.8) (best results when the line from sun to moon points to the upper right)

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#5. Use the animate function in the plots package of Maple to illustrate the earth moving around the sun (where the orbit is drawn, but a 
#point labeled E is moving around the sun). 
#If possible also have the moon going around the earth. The input parameters should be an abitrary year length (365.5 for us) and lunar 
#sideral period (27.32 for us), as well as the center of the sun, and the distance of the sun to the earth. You can assume that it is a 
#circular orbit.

#Skipped