The two idealized curves on the accompanying graph show:
 Solid curve: the daily average intensity of solar radiation absorbed
by the earth's surface at different times of the year at Sacramento, California
(about 38°N latitude);

Dashed curve: the daily average net intensity
of terrestrial radiation emitted by the earth's surface at different
times of the year at Sacramento. ("Net" in this case means the difference
between terrestrial radiation emitted and terrestrial radiation absorbed.
The terrestrial radiation that is absorbed at the surface comes from the
atmosphere, which emits terrestrial radiation downward as well as upward
to space. To simplify things, we'll assume in this exercise that the terrestrial
radiation emitted downward by the atmosphere is constant during the year, so the net emission varies only because emission does.)
The beginnings of individual months of the year are numbered along the horizontal
axis of the graph, starting arbitrarily with 0 (which isn't necessarily January).
For the purposes of this assignment, assume that the earth at Sacramento is
gaining and losing heat (on the average) only by absorbing and emitting radiation
(a simplification that is only partly true but is good enough to get some interesting
results!).
Below is a list of events that occur at either (a) specific times of year or (b) ranges of times during
the year. The specific time of year, or range of times, when each event occurs can be determined using information on the graph. For each
event on the list, determine the approximate time of the year (for example, "late June",
or "mid April") or the range of times during the year (for example,
"late June to late December") when the event occurs. (You
need not give answers within any particular month more precise than "early",
"late", or "mid" month; the graph doesn't permit answers more precise than that.
You should be able to determine which month is which, so refer to months
by name rather than by number.)
Depending on the question, you might be able to determine each answer by
(1) simply reading the graph directly; (2) applying common sense and/or information
that you've already learned this semester; or (3) applying one or two of
the following three principles to interpret information on the graph:
 A basic law of radiation, the StefanBoltzmann relation: The warmer most objects/materials are, the more intensely they emit radiative energy.
 An object's temperature will be increasing if it is gaining heat
faster than it is losing it; it's temperature will be decreasing
if it is losing heat faster than it is gaining it; and it's temperature
won't be changing if it is gaining and losing heat at the same rate.
 The bigger the net rate of heat gain or loss is (that is, the bigger
the difference between the rates at which an object simultaneously
gains and loses energy by separate mechanisms), the faster the temperature
will change.
Note that the latter two principles above really just follow directly from a simplified,
semigeneral statement of the conservation of energy applied to the heat budget for the earth's surface:
Rate at which an object's temperature changes

∝ (i.e., is proportional
to)

Rate at which
the object's heat content changes

=

Rate at which
the object absorbs
solar radiative energy

— 
Net rate at which
the object emits
radiative energy 
For events 4 through 7 below, there there are two different ways
to get an answer. Each of the two ways of answering uses (directly
or indirectly) one of the three physical principles listed above. For one way
of answering, you'll need information from only one of the two curves
on the graph. For the other way of answering, you'll need information from both
curves. (Hint: If you read the principles above closely, you'll note that the two
that are based on the conservation of energy require information from both curves.)
You'll find that using information from both curves generally gives you a more
precise answer than using information from only one curve.
For events 4 through 7, identify the time of year or range of time of year when each occurs, using both ways of answering the question.
(Of course, since the results of both approaches should agree, this happens
to provide a way to check your answers.)
For each event (1 through 7), briefly summarize how you got your answers.
For those questions (that is, questions 4 through 7) for which it is possible
to answer in two different ways (by applying different physical principles),
your summary should include how you applied each principle.
 The winter and summer solstices and the spring and autumn equinoxes (based
in part on the graph, not based solely on prior knowledge of the date)
 All else being equal, the dailyaverage intensity of absorption of solar
radiation would be greatest on the summer solstice because that is when
both
the sun angle is greatest and the day is longest. The peak of the solar absorption
curve lies near the middle of or late in the month labeled "7".
In the Northern Hemisphere the summer solstice occurs in June, so this
would
be in midJune or lateJune. (Of course, we actually know that the precise
date is usually June 21 or 22.)
 All else being equal, the dailyaverage intensity of solar absorption
would be least on the winter solstice because that is when both the sun angle
is least and the day is shortest. The minimum in the solar absorption curve
lies in late December (that is, mid to late in month #s 1 and 13 on the graph,
counting months backward and forward from the month of the summer solstice).
 The spring equinox lies mid or late March (months #4 and #16). This lies
almost exactly three months after the winter solstice and three months before the
summer solstice, when the solar absorption is increasing (in fact, it's increasing
the most rapidly at that time, all else being equal).
 Mid or late September (month #10), by similar reasoning as for the fall
equinox.
 Time when solar radiation is the least intense
Mid or late December, when the solar absorption curve is at a minimum.
(Of course, by definition this occurs on the day of the winter solstice.)
 Time when IR emission is the most intense
Late July (month #8) or early August (month #9), the apparent
peak of the emission curve.
 The single warmest or coldest time of the year (choose one)
Warmest time of year:
(a) Late July (month #8) or early August (month #9). The StefanBoltzmann
law of radiation connects emission to temperature, so the time when emission
is greatest must
be the
time when temperature is greatest.
(b) Late July. Before the warmest time of year the temperature must be increasing,
and afterwards it must be decreasing. (This is the nature of the maximum value
of any quantity.) In this case, according to the heat budget equation, for the temperature to be increasing absorption must exceed emission, and
for temperature to be decreasing emission must exceed absorption. Hence, the absorption curve must lie above the emission
curve leading up to the single warmest time of year, and the opposite must be true immediately after the warmest time. It follows that
the warmest time of year must be at the precise moment when the two curves
cross.
Coldest time of year:
(a) Early or mid February (months #3 and 15), based on similar reasoning as
above.
(b) Very early February, based on similar reasoning as above.
 Range of times when temperature is increasing or the
range of times when it is decreasing (choose one)
Increasing temperature:
(a) The temperature must be increasing from its minimum until its maximum.
Based solely on the emission curve , which the StefanBoltzmann
law of radiation relates to temperature, that period
appears to be roughly from early or mid February to late July or early August.
(b) According to the heat budget equation in this case, the temperature
will be increasing when absorption exceeds emission. This occurs from early
February to late July, when the absorption curve lies above the emission curve.
Decreasing temperature:
(a) Based reasoning similar to that above, this would occur from
late July or early August until early or mid February.
(b) Based on reasoning similar to that above, this would occur from late July
to early February.
 Single time (approximately) when temperature is increasing the fastest
(a) Based on the emission curve and the StefanBoltzmann
law of radiation only,
this occurs when the slope of the emission curve is greatest (since the temperature
would be increasing the fastest where the emission is increasing the fastest).
This is not easy to tell from the emission curve very precisely, but it looks
like it occurs somewhere in late April or May.
(b) According to the heat budget equation in this case, the temperature
should increase the fastest where the difference between absorption and emission
is the greatest. By marking the (vertical) distance between the absorption
and emission curves on a piece of paper, the maximum separation between the
curves appears to be in late April or early May.
 A time when (avg. daily) temperature isn't changing (that is, isn't increasing
or decreasing) (if there is more than one, choose one)
(a) The temperature must stop changing at the times when it is maximum
or minimum. Before each such point the temperature changes in one sense (that
is, with one sign, positive or negative) and afterwards changes in the other
sense, so the temperature goes from increasing to decreasing or vice versa
at maximum and minimum points. To do this (that is, change the sense with
which it is changing), the temperature must momentarily stop changing at the
maximum and minimum points. The answers were given in (4)(a).
(b) According to the heat budget equation in this case, the temperature
should stop changing when absorption equals emission, which happens when the
curves cross. The answers were given in (4)(b).
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