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Notebook[{
Cell[CellGroupData[{
Cell["Quirks of Time ", "Title"],
Cell["(*Authors to supply subtitle with edits.*)", "Text"],
Cell["by L. R. King and Todd G. Will", "Text"],
Cell[BoxData[""], "Input"],
Cell[CellGroupData[{
Cell["Introduction.", "Section"],
Cell[TextData[{
"The day with the fewest number of daylight hours, sometimes referred to as \
the shortest day of the year, is winter solstice, which occurs in the \
northern hemisphere on December 21st or 22nd. Given that the number of \
daylight hours increases after the winter solstice one should naturally \
expect that the sun would rise earlier on subsequent days. Surprisingly \
this is not the case. Depending on latitude, the sun continues to rise later \
each day for ",
StyleBox["as much as",
FontVariations->{"CompatibilityType"->0}],
" several weeks past winter solstice despite the fact that the number of \
daylight hours steadily increases. We refer to this curious phenomenon as \
Delayed Early Sunrise (DES) and offer an explanation for it based solely on \
the fact that the earth's polar axis is tilted, making an approximate angle \
of 23.5\[Degree] with its orbital plane. "
}], "SmallText"],
Cell[TextData[{
"The explanation comes in two parts. First, we establish the connection \
between the time of sunrise and the time of high noon. By ",
"high noon",
" we mean the time when the sun is highest in the sky. Second, we \
establish the connection between high noon and twelve o'clock noon. ",
StyleBox["It is the discrepancy between high noon and twelve o'clock noon \
(quirk of time!) that accounts for delayed early sunrise in our model.",
FontVariations->{"CompatibilityType"->0}]
}], "SmallText"]
}, Open ]],
Cell[CellGroupData[{
Cell["Choosing coordinates", "Section"],
Cell["\<\
We choose a coordinate system fixed to the stars with earth at the \
origin and the z-axis aligned with the north pole.\
\>", "SmallText"],
Cell[BoxData[{
\(\(SetOptions[
Graphics3D, \[IndentingNewLine]PlotRange \[Rule] {{\(-3\),
3}, {\(-3\), 3}, {\(-2\),
2}}, \[IndentingNewLine]LightSources \[Rule] {}, \
\[IndentingNewLine]Boxed \[Rule]
False, \[IndentingNewLine]AmbientLight \[Rule]
White, \[IndentingNewLine]ViewPoint \[Rule] {4, 0,
1}, \[IndentingNewLine]BoxRatios \[Rule]
Automatic];\)\[IndentingNewLine]\), "\[IndentingNewLine]",
\(\(Show[earth, equatorialPlane, xyzAxes,
PlotRange \[Rule] All];\)\)}], "Input"],
Cell[TextData[{
"In this coordinate system the sun orbits the earth counter-clockwise as \
viewed from the north pole. To keep the graphics uncomplicated we depict a \
12 day year with time ",
Cell[BoxData[
\(TraditionalForm\`t = 0\)]],
" corresponding to twelve o'clock noon on day 0. As a further \
simplification we initially ignore the tilt of the orbital plane and show the \
sun orbiting in the equatorial plane. "
}], "SmallText"],
Cell[BoxData[{
\(Clear[sun, days]\), "\n",
\(\(sun[t_, days_: 365.25, tilt_: 23.5 Degree] = \
RotationMatrix3D[\(-Pi\)/2, tilt,
Pi/2] . {Cos[\(\(2 Pi\)\/days\) t], Sin[\(\(2 Pi\)\/days\) t],
0};\)\n\), "\n",
\(Clear[sunOrbit]\), "\n",
\(sunOrbit[t_, days_: 365.25, tilt_: 23.5 Degree,
col_: 0.3] := \[IndentingNewLine]{orbitalPlane[tilt,
col], (*Defined\ in\ initialization\ \
*) \[IndentingNewLine]Graphics3D[{Hue[col, 1,
0.6], \[IndentingNewLine]dateLabel[t, days,
tilt], \ (*Defined\ in\ initialization\ *) \[IndentingNewLine]\
\({PointSize[0.01], Point[#], PointSize[0.02], Point[2.7 #],
Line[{#, 2.7 #}]} &\) /@ \ \ Table[
sun[s, days, tilt], {s, 0, t,
days/12}]}]\[IndentingNewLine]}\[IndentingNewLine]\), "\
\[IndentingNewLine]",
\(noonMeridian[t_,
days_] := \[IndentingNewLine]Graphics3D[{Red, Thickness[0.003],
Line[Table[
1.01 RotationMatrix3D[\(\(-2\) \[Pi]\ t\)\/days, 0,
0] . {Cos[Pi\ s], 0, Sin[Pi\ s]}, {s, \(-1\)/2, 1/2,
0.1}]]\[IndentingNewLine]}]\[IndentingNewLine]\), "\n",
\(<< Graphics`Animation`\), "\n",
\(\(SetOptions[Animate, Closed \[Rule] True,
Frames \[Rule] 12];\)\), "\n",
\(\(Animate[{\ \ earth, sunOrbit[t, 12, 0, 0], noonMeridian[t, 12]}, {t,
0, 12}];\)\)}], "Input"],
Cell[TextData[{
"In addition to the sun's orbit about the earth, the animation gives a \
potentially misleading picture of the rotation of the earth on its axis. Any \
confusion about this motion can be cleared up by focusing on the earth's line \
of longitude shown in red. We refer to this meridian as the ",
StyleBox["twelve o'clock meridian",
FontWeight->"Bold"],
" since the animation shows its location at twelve o'clock noon on each \
day. "
}], "SmallText"],
Cell[TextData[{
"Note that at twelve o'clock noon on day 0 when ",
Cell[BoxData[
\(TraditionalForm\`t = 0\)]],
", the sun appears highest in the sky to folks living on the twelve o'clock \
meridian. During the first day, as the sun proceeds through 1/12 of its \
orbit, the animation seems to show the earth making 1/12 of a revolution on \
its axis. In fact, during the first day the earth rotates counter-clockwise \
on its axis 13/12 of a revolution causing the sun to set in the west, rise in \
the east and again appear directly overhead to viewers on the twelve o'clock \
meridian at twelve o'clock noon on day 1. In the next 24 hours the earth \
rotates another ",
Cell[BoxData[
\(TraditionalForm\`13/12 = 1 + 1\/12\)]],
" of revolution, the extra 1/12 of a revolution again allowing the twelve \
o'clock meridian to catch up to the new location of the sun at twelve o'clock \
noon."
}], "SmallText"],
Cell[TextData[{
"Tracking the twelve o'clock meridian gives us a connection between clock \
time and astronomical reality. In our currently simplified model, in which \
the sun orbits in the equatorial plane, clock time also appears to be \
connected to the orbit of the sun. In particular we note in this untilted \
model that throughout the year viewers on the twelve o'clock meridian \
experience high noon at twelve o'clock noon. For this reason we refer to the \
sun in this simplified model as the \"clock-sun\". ",
StyleBox[" ",
FontSlant->"Plain",
FontVariations->{"CompatibilityType"->0}],
StyleBox["In fact, the clock sun for a normal 365.25 day year gives the \
correct time if we choose the 12 o'clock meridian to be the prime meridian (0 \
degrees longitude).",
FontVariations->{"CompatibilityType"->0}],
" We will see shortly that orbital tilt complicates the relationship \
between high noon and twelve o'clock noon,",
StyleBox[" ",
FontSlant->"Plain",
FontVariations->{"CompatibilityType"->0}],
StyleBox["introducing what we call the quirk of time",
FontVariations->{"CompatibilityType"->0}],
"."
}], "SmallText"],
Cell["\<\
Now look at the more realistic orbit of the sun incorporating the \
23.5\[Degree] tilt of the orbital plane.\
\>", "SmallText"],
Cell[BoxData[
\(\(Animate[{\ earth, equatorialPlane, \ sunOrbit[t, 12]}, {t, 0,
12}];\)\)], "Input"],
Cell[TextData[{
"The orbital plane has been tipped",
StyleBox[" ",
FontSlant->"Plain",
FontVariations->{"CompatibilityType"->0}],
StyleBox["along the axis defined by the equinoxes",
FontVariations->{"CompatibilityType"->0}],
" so that winter solstice occurs at twelve o'clock noon on day 0. This puts \
the summer solstice on day 6 and the spring and fall equinoxes on days 3 and \
9, respectively. Note how the ",
StyleBox["declination",
FontWeight->"Bold"],
" of the sun, that is the angle the sun makes with the equatorial plane, \
varies from minimum of ",
Cell[BoxData[
FormBox[
RowBox[{
FormBox[\(-23.5\),
"TraditionalForm"], "\[Degree]"}], TraditionalForm]]],
" ",
StyleBox["at",
FontVariations->{"CompatibilityType"->0}],
" winter solstice to a maximum of ",
Cell[BoxData[
\(TraditionalForm\`23.5 \[Degree]\)]],
" ",
StyleBox["at",
FontVariations->{"CompatibilityType"->0}],
" summer solstice. It is this seasonal variation that makes sunrise times \
latitude dependent. "
}], "SmallText"],
Cell["\<\
Here is a look at how the sun illuminates the earth throughout the \
year.\
\>", "SmallText"],
Cell[CellGroupData[{
Cell[BoxData[
\(\(litEarth[t_, days_: 365.25, tilt_: 23.5 Degree] :=
Module[{sunDir = sun[t, days, tilt], n1,
mat}, \[IndentingNewLine]n1 =
Cross[sunDir, {0, 0, 1.0}]; \[IndentingNewLine]n1 =
n1/Sqrt[n1 . n1]; \[IndentingNewLine]mat =
Transpose[{n1, Cross[n1, sunDir],
sunDir}]; \[IndentingNewLine]{halfLight /. {x__Real} \
\[RuleDelayed] mat . {x}, \ \ (*\
halfLight\ defined\ in\ initialization\ \
*) \[IndentingNewLine]WireFrame[
Sphere[1, 12, 16]]\[IndentingNewLine]}];\)\)], "Input"],
Cell[BoxData[
\(Animate[{\ equatorialPlane, litEarth[t, 12], \
sunOrbit[t, 12]}, \[IndentingNewLine]{t, 0, 12}]\)], "Input"]
}, Closed]],
Cell["\<\
Here's the same animation viewed looking down from the north \
pole.\
\>", "SmallText"],
Cell[BoxData[
\(Animate[{\ equatorialPlane, litEarth[t, 12], \
sunOrbit[t, 12]}, \[IndentingNewLine]{t, 0,
12}, \[IndentingNewLine]ViewPoint \[Rule] {0, 0,
5}, \[IndentingNewLine]ViewVertical \[Rule] {\(-1\), 0,
0}]\)], "Input",
AnimationDisplayTime->0.2197],
Cell["\<\
The animation above shows seasonal variations in the lighting of \
the earth, NOT the daily variations of night and day. Night and day are \
caused by the undepicted rotation of the earth on its axis. \
\>", \
"SmallText"],
Cell[TextData[{
"One can see that at northern latitudes the fewest hours of daylight occur \
on the day of winter solstice and the greatest number on the day of summer \
solstice. Consequently the angle from the sun to the location of sunrise \
(the boundary of blue and yellow) is smallest ",
StyleBox["at",
FontVariations->{"CompatibilityType"->0}],
" winter solstice and greatest ",
StyleBox["at",
FontVariations->{"CompatibilityType"->0}],
" summer solstice. Since the earth rotates on its axis at a constant rate, \
focusing on this angle will provide a connection between sunrise time and the \
time of high noon. To compute this angle we first need to compute the \
declination of the sun."
}], "SmallText"],
Cell[TextData[{
"To find the declination at any time t first note that our parametrization \
for the sun can be viewed as a unit vector. Thus the cosine of the angle \
between the sun and the north pole can be computed as the dot product of the \
sun vector and <0,0,1>. Subtracting this angle from ",
Cell[BoxData[
\(TraditionalForm\`\[Pi]\/2\)]],
" gives the declination for any time t, ",
StyleBox["declination[t]",
FontSlant->"Plain",
FontVariations->{"CompatibilityType"->0}],
"."
}], "SmallText"],
Cell[BoxData[
\(declination[t_, days_: 365.25, tilt_: 23.5 Degree] =
FullSimplify[\
Pi/2 - ArcCos[sun[t, days, tilt] . {0, 0, 1}]]\)], "Input"]
}, Open ]],
Cell[CellGroupData[{
Cell["From sunrise to high noon.", "Section"],
Cell[TextData[{
"As mentioned above, the connection between high noon and sunrise will be \
made by focusing on the angle that the earth rotates through from sunrise to \
high noon. We call this angle ",
Cell[BoxData[
\(TraditionalForm\`\[Rho]\)]],
" and note that it depends on both the latitude \[Lambda] and the \
declination of the sun which is determined by time. The following \
illustrates the angle \[Rho] at winter solstice for latitude \[Lambda]\
\[TildeTilde]25\[Degree] N. "
}], "SmallText"],
Cell[BoxData[
\(\(Show[\[IndentingNewLine]litEarth[0,
12], \[IndentingNewLine]Graphics3D[{Red,
Thickness[
0.005], \[IndentingNewLine]Text["\<\[Rho]\>", {0.2 Cos[0.5],
0.2 Sin[0.5], 1}], \[IndentingNewLine]Line[
Join[{{0, 0, 1}},
Table[{0.7 Cos[t], 0.7 Sin[t], 1}, {t, 0, 1.16,
1.16/6}], {{0, 0,
1}}]]\[IndentingNewLine]}], \[IndentingNewLine]ViewPoint \
\[Rule] {0, 0, 5}, \[IndentingNewLine]ViewVertical \[Rule] {\(-1\), 0,
0}, \[IndentingNewLine]PlotRange \[Rule] All];\)\)], "Input"],
Cell[TextData[{
"In the previous section we developed a function, declination[t], to \
compute the declination of the sun at any time t. For simplicity we fix the \
sun in the xz-plane and use the declination function to create a unit vector \
that points to the sun with the correct declination at time ",
Cell[BoxData[
\(TraditionalForm\`t\)]],
": \n\t{Cos[declination[t]], 0, Sin[declination[t]]}.\n\t\nIn our \
coordinate system the points on earth at latitude \[Lambda] are swept out by \
the vector \n \t{Cos[\[Theta]]Cos[\[Lambda]],Sin[\[Theta]]Cos[\[Lambda]],Sin[\
\[Lambda]]} as \[Theta] ranges from 0 to 2\[Pi].\n\nSunrise and sunset occur \
at this latitude at the values of \[Theta] for which these two vectors are \
perpendicular. ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" finds both of these values, the second of which corresponds to sunrise."
}], "SmallText"],
Cell[BoxData[{
\(Clear[days, tilt]\), "\[IndentingNewLine]",
\(\[Rho][t_, \[Lambda]_, days_: 365.25, tilt_: 23.5 Degree] =
FullSimplify[\(Solve[
0 == {Cos[\[Theta]] Cos[\[Lambda]], Sin[\[Theta]] Cos[\[Lambda]],
Sin[\[Lambda]]} . {Cos[declination[t, days, tilt]], 0,
Sin[declination[t, days,
tilt]]}, \[Theta]]\)\[IndentingNewLine][\([2, 1,
2]\)]\ ]\)}], "Input"],
Cell[TextData[{
"Given that the earth rotates on its axis roughly once every 24 hours we \
have that sunrise on day t should occur roughly ",
Cell[BoxData[
\(TraditionalForm\`\(24\/\(2 \[Pi]\)\) \[Rho][t, \[Lambda]]\)]],
" hours before high noon on day t. "
}], "SmallText"],
Cell[TextData[{
"There are two small errors in our reasoning. First, given our coordinate \
system, the earth rotates 2\[Pi] ",
Cell[BoxData[
\(TraditionalForm\`\(days + 1\)\/days\)]],
" radians each 24 hours ",
StyleBox["(days = number of days in a year)",
FontVariations->{"CompatibilityType"->0}],
". Second, the declination of the sun at high noon, which we use to \
predict sunrise time, differs slightly from the declination of the sun at \
sunrise, which occurred roughly six hours prior to high noon. Although these \
errors would be significant for the 12 day year we display in our graphics, \
they are both negligible in the calculations we perform for the real year \
with 365 days."
}], "SmallText"]
}, Open ]],
Cell[CellGroupData[{
Cell["From high noon to twelve o'clock noon.", "Section"],
Cell[TextData[{
"We saw above that sunrise at latitude \[Lambda] on day t occurs roughly ",
Cell[BoxData[
\(\(12\/\[Pi]\) \[Rho][t, \[Lambda]]\)]],
" hours before high noon. Here's a plot for our 365 day year showing \
sunrise times at 31\[Degree] N latitude assuming that high noon occurs at \
twelve o'clock noon. "
}], "SmallText"],
Cell[BoxData[
\(\(Plot[\
12 - \(12\/\[Pi]\) \[Rho][t, 31 Degree], {t, \(-200\),
200}];\)\)], "Input"],
Cell[TextData[{
"Recalling that day ",
Cell[BoxData[
\(TraditionalForm\`t = 0\)]],
" corresponds to the winter solstice we see that there is no Delayed Early \
Sunrise. That is, the latest sunrise occurs on the winter solstice at \
approximately 7 a.m. and earlier sunrises start immediately on the following \
days. The reality of DES forces the conclusion that high noon and twelve \
o'clock noon do not always coincide. "
}], "SmallText"],
Cell["\<\
As stated in the beginning our explanation for DES rests on the \
tilt of the orbital plane. To see the effect of the tilt look at the \
following plot which depicts the orbit of the sun with and without the \
orbital tilt.\
\>", "SmallText"],
Cell[BoxData[
\(\(Show[
earth, \[IndentingNewLine]sunOrbit[11, 12, 0,
0], \[IndentingNewLine]sunOrbit[11, 12, 40 Degree,
0.3]\[IndentingNewLine]];\)\)], "Input"],
Cell["\<\
The red graphics show a 12 day orbit with no orbital tilt. The \
green graphics show a 12 day orbit with an exaggerated orbital tilt of 40\
\[Degree]. Look at the same plot from above the north pole.\
\>", "SmallText"],
Cell[BoxData[
\(\(Show[
earth, \[IndentingNewLine]sunOrbit[11, 12, 0,
0], \[IndentingNewLine]sunOrbit[11, 12, 40 Degree,
0.3], \[IndentingNewLine]ViewPoint \[Rule] {0, 0,
5}, \[IndentingNewLine]ViewVertical \[Rule] {\(-1\), 0,
0}];\)\)], "Input"],
Cell["\<\
Recall that the labels 0 through 11 mark the location of the sun at \
twelve o'clock noon on each day. The red graphics depict the fictitious \
clock-sun which orbits the earth in the equatorial plane. The red lines \
depict the positions of the twelve o'clock meridian. Recall that at twelve \
o'clock noon on each day the location of the clock-sun coincides with the \
location of twelve o'clock meridian. \
\>", "SmallText"],
Cell[TextData[{
"The green graphics show an effect of an exaggerated orbital tilt of 40\
\[Degree]. ",
StyleBox["The green lines depict the positions of lines of longitude along \
which viewers experience high noon. The mis-match of the two sets of lines \
is evident in the graphic.",
FontVariations->{"CompatibilityType"->0}],
" At time ",
Cell[BoxData[
\(TraditionalForm\`t = 0\)]],
" both the red sun and green sun are at high noon for viewers on the twelve \
o'clock meridian. In 24 hours at twelve o'clock noon on day 1 the twelve \
o'clock meridian has made 13/12 of a revolution to once again align with the \
fictitious clock-sun. But at this time the green sun is not overhead. The \
earth must continue its counter-clockwise rotation for several more minutes \
before the green sun is overhead. Thus on the days following winter solstice \
high noon occurs after twelve o'clock noon. The graphic shows that the same \
behavior continues until the spring equinox when once again high noon and \
twelve o'clock noon coincide. "
}], "SmallText"],
Cell["\<\
At twelve o'clock noon on day 4 the folks on the twelve o'clock \
meridian have rotated past the green sun and thus past the position of high \
noon. Thus on days 4 and 5 high noon occurs before twelve o'clock noon. \
Days 7 and 8 following summer solstice are similar to those following winter \
solstice in that high noon again occurs after twelve o'clock noon. On days \
10 and 11 following the spring equinox high noon again occurs before twelve \
o'clock noon. \
\>", "SmallText"],
Cell["\<\
To measure the time between high noon and twelve o'clock noon we \
will measure the angle the earth rotates to reach one from the other. We \
refer to this angle as quirk[t] and to measure it we begin by projecting the \
location of the tilted sun onto the equatorial plane and then \
normalizing.\
\>", "SmallText"],
Cell[BoxData[{\(<< LinearAlgebra`Orthogonalization`\), "\[IndentingNewLine]",
RowBox[{\(projectedSun[t_, days_, tilt_]\), "=",
RowBox[{"Normalize", "[", " ",
RowBox[{
RowBox[{"(", GridBox[{
{"1", "0", "0"},
{"0", "1", "0"},
{"0", "0", "0"}
}], ")"}], ".", \(sun[t, days, tilt]\)}], "]"}]}]}], "Input"],
Cell[TextData[{
"The direction of the twelve o'clock meridian on day t is given by ",
Cell[BoxData[
\(TraditionalForm\`{Cos[\ \(2\ \[Pi]\ t\)\/days],
Sin[\(2\ \[Pi]\ t\)\/days\ ], 0}\)]],
". We could get the unsigned angle between high noon and the twelve \
o'clock meridian by computing the ArcCosine of these two vectors. However \
the following trick of adding and subtracting ",
Cell[BoxData[
\(TraditionalForm\`\[Pi]\/2\)]],
" will give us the signed angle that we need."
}], "SmallText"],
Cell[BoxData[
\(quirk[t_, days_: 365.25, tilt_: 23.5 Degree] =
FullSimplify[\[IndentingNewLine]\[Pi]\/2 -
ArcCos[{Cos[\(2\ \[Pi]\ t\)\/days + \[Pi]\/2],
Sin[\(2\ \[Pi]\ t\)\/days + \[Pi]\/2], 0} . \
projectedSun[t, days, tilt]]\[IndentingNewLine]]\)], "Input"],
Cell["\<\
Again relying on the constant rate of the earth's rotation we plot \
the minutes from twelve o'clock noon to high noon on the days following the \
winter solstice.\
\>", "SmallText"],
Cell[BoxData[
\(\(Plot[\(\(60\ 24\)\/\(2 \[Pi]\)\) quirk[t], {t, 0, 365.25},
PlotLabel \[Rule] "\"];\)\)], \
"Input"],
Cell[TextData[{
"Finally we see the root cause for DES. The tilt of the orbital plane \
causes the sun to be increasingly tardy relative to the clock for nearly 50 \
days following the winter solstice. To see the effect on sunrise we update \
our previous formula, ",
Cell[BoxData[
\(TraditionalForm\`sunrise[t, \[Lambda]] =
highnoon[t]\ - \(\(12\/\[Pi]\) \(\[Rho][t, \[Lambda]]\)\(\ \)\)\)]],
", replacing highnoon with ",
Cell[BoxData[
\(TraditionalForm\`12 + \(\(12\/\[Pi]\) \(quirk[t]\)\(\ \)\)\)]]
}], "SmallText"],
Cell[BoxData[{
\(Clear[sunrise]\), "\[IndentingNewLine]",
\(\(sunrise[t_, \[Lambda]_, days_: 365.25,
tilt_: 23.5 Degree] = \[IndentingNewLine]12 + \(12\/\[Pi]\)
quirk[t, days, tilt]\ - \(12\/\[Pi]\) \[Rho][t, \[Lambda], days,
tilt];\)\)}], "Input"],
Cell["\<\
Here is a plot showing sunrise times for various latitude on the \
days following winter solstice.\
\>", "SmallText"],
Cell[BoxData[
\(\(Show[\[IndentingNewLine]GraphicsArray[\[IndentingNewLine]Partition[\
\[IndentingNewLine]Table[\[IndentingNewLine]Plot[
sunrise[t, \[Lambda]\ Degree], {t, \(-10\),
50}, \[IndentingNewLine]DisplayFunction \[Rule]
Identity, \[IndentingNewLine]PlotLabel \[Rule] "\" <> ToString[\[Lambda]]], \[IndentingNewLine]{\[Lambda], 0, 60,
12}], \[IndentingNewLine]3]\[IndentingNewLine]],
DisplayFunction \[Rule] $DisplayFunction\[IndentingNewLine]];\)\)], \
"Input"],
Cell[TextData[{
"We have seen that after winter solstice both quirk[t] and \[Rho][t,\
\[Lambda]] increase. Thus the change in \n\t\t",
Cell[BoxData[
\(TraditionalForm\`sunrise[t] =
12 + \(12\/\[Pi]\)
quirk[t]\ \ - \(\(12\/\[Pi]\) \(\[Rho][
t, \[Lambda]]\)\(\ \)\)\)]],
" \nis determined by the relative change in quirk and \[Rho]. \nAs long as \
the increase in quirk is larger than the increase in \[Rho], sunrise time \
continues to be later after winter solstice. The delay in earlier sunrise is \
greatest at the equator where \[Rho] doesn't change at all. Because the \
increase in \[Rho] becomes more pronounced with greater latitude, the delayed \
early sunrise is of shorter duration for higher latitudes. "
}], "SmallText"]
}, Open ]],
Cell[CellGroupData[{
Cell["About Our Model.", "Section"],
Cell["\<\
Our analysis of delayed early sunrise, concentrating as is does on \
the tilt of the earth , omits consideration of the earth's non-uniform motion \
along its elliptical orbit. The earth is moving at its fastest in early \
January when it is closest to the sun (perihelion), gradually slowing until \
early July when it is most distant from the sun (aphelion) at wich time the \
process is reversed. At perihelion the earth's rotation needs a little more \
time to catch up to the sun, tending to make the sun late; at apehelion just \
the opposite happens. Because aphelion and perihelion are off-set from the \
solstices, the two major influences on quirk - non-uniform motion and tilt - \
are accordingly off-set. The result is the nice symmetry of our plot of \
quirk[t] is lost. Moreover, the \"real\" plot of quirk[t] differs from ours \
in that the amplitude is greater (about 15 versus 10 minutes). Other (finer) \
points: we set our clock at 12:00 noon precisely at winter solstice for \
computational convenience and played the role of an observer at 0 degrees \
longitude (12 o'clock meridian), avoiding corrections due to civil time \
zones. \
\>", "SmallText"]
}, Open ]],
Cell["Graphics Primitives", "Section"],
Cell[BoxData[{
\(<< Graphics`Shapes`\), "\n",
\(<< Graphics`Colors`\n\), "\[IndentingNewLine]",
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"\nWebsites\nU. S. Naval Observatory Data Services: \t\
http://aa.usno.navy.mil/AA/data/\nThe Analemma:\t\t\t\t\
http://www.analemma.com/"
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L. R. King and Todd G. Will
Department of Mathematics
Davidson College
Box 1719
Davidson NC 28036\
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