Chen C.J. Physics of Solar Energy

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92  Tracking Sunlight

Figure 4.10 Daily solar radiation energy on a latitude-tilt surface. Solar panels placed on

a latitude-tilt surface enjoy almost the maximum solar radiation over the entire year.

only limited by the angle of incidence,

cos δ



π/2

−π/2

cos ωdω

cos δ.

(4.53)

As shown in Fig. 4.10, in this period of the year, the daily radiation energy is inde-

pendent of latitude. However, between the autumnal equinox and the vernal equinox

of the next year, sunrise is later than 6:00 am and sunset is earlier than 6:00 pm. The

daily radiation energy is

cos δ



−ω

cos ωdω

cos δ sin ω

(4.54)

Also shown in Fig 4.10, the radiation energy is reduced and depends on the latitude of

the location, but not by much. Over the entire year, maximum solar radiation energy

is obtained.

If the surface is allowed to follow the apparent motion of the Sun, the daily radiation

energy can be further enhanced. Consider a solar panel mounted on an axis parallel to

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4.3 Treatment in Solar Time  93

Figure 4.11 Daily solar radiation energy on a surface with tracking. Daily radiation received

by a solar panel mounted on an axis parallel to the axis of Earth and the rotates uniformly one turn per

day, similar to that on Plate 7. The average daily solar radiation over the entire year is independent

of latitude, see Table 4.2.

the axis of Earth and that rotates uniformly one turn per day. The daily solar radiation

energy is

cos δω

. (4.55)

As shown in Fig. 4.11, the daily solar radiation on a surface with single-axis tracking

is substantially higher than for ﬁxed surfaces. The advantage is more apparent in the

average direct daily solar radiation

over a year, as shown in Table 4.2. The solar

radiation on a surface with tracking is more than 50% higher than all ﬁxed surfaces.

Nevertheless, because of the eﬀect of shadows, it does not save horizontal area.

Table 4.2: Average Daily Solar Radiation on Various Surfaces

in kWh/day at Latitude 30

◦

Vertical, facing south 3.72 4.57 5.25 5.60

Vertical, facing west or east 3.31 2.93 2.46 1.91

Horizontal 6.25 5.43 4.39 3.11

Latitude tilt facing south 7.27 7.21 7.08 6.77

Optimum single-axis tracking 11.50 11.50 11.50 11.50

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94  Tracking Sunlight

4.3.5 The 24 Solar Terms

As shown in previous sections, the motion of the Sun in a calendar year is determined

by the Sun’s ecliptic longitude l. The vernal equinox at l =0

◦

or 360

◦

, the summer

solstice at l =90

◦

, the autumnal equinox at l = 180

◦

, and the winter solstice at

l = 270

◦

are the four cardinal points. To study the position of the Sun over a year,

more points are needed. In traditional Western calendars, including the Julian calendar

and the Gregorian calendar, the deﬁnitions of the 12 months are not based on natural

phenomena. The calendars are not accurately synchronized with the motion of the

Sun. Therefore, it is inaccurate and inconvenient to use calendar dates to specify the

motion of the Sun.

In East Asia, however, for over two thousand years, a purely solar-based calendar

system has been used: the 24 solar terms. It is deﬁned solely on the basis of the orbital

motion of Earth around the Sun. Similar to dividing a mean solar day into 24 h,

Table 4.3: The 24 Solar Terms

No. Name In Pinyin l (deg) Approx. date

0 Winter solstice D¯ongzh`ı 270

◦

December 22

1 Minor cold Xi˘aoh´an 285

◦

January 6

2Majorcold D`ah´an 300

◦

January 20

3 Spring commences L`ıch¯un 315

◦

February 4

4 Rain water Y˘ush˘ui 330

◦

February 19

5 Insect awakes J¯ıngzh´e 345

◦

March 6

6 Vernal equinox Ch¯unf¯en 0

◦

March 21

7 Pure brightness Q¯ıngm´ıng 15

◦

April 5

8 Grain rain G˘uy˘u30

◦

April 20

9 Summer commences L`ıxi`a45

◦

May 6

10 Grain forms Xi˘aom˘an 60

◦

May 21

11 Grain in ear M´angzh`ong 75

◦

June 6

12 Summer solstice Xi`azh`ı90

◦

June 21

13 Minor heat Xi˘aosh˘u 105

◦

July 7

14 Major heat D`ash˘u 120

◦

July 23

15 Autumn commences L`ıqi¯u 135

◦

August 8

16 Heat recedes Ch˘ush˘u 150

◦

August 23

17 White dew B´ail˘u 165

◦

September 8

18 Autumnal equinox Qi¯uf¯en 180

◦

September 23

19 Cold dew H´anl˘u 195

◦

October 8

20 Frost descents Shu¯angji`ang 210

◦

October 23

21 Winter commences L`ıd¯ong 225

◦

November 7

22 Minor snow Xi˘aoxu˘e 240

◦

November 22

23 Major snow D`axu˘e 255

◦

December 7

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the solar term system divides a solar year (the time between two consecutive vernal

equinoxes) into 24 equal parts; see Table 4.3. Each year, the exact date and time of

day of each of the 24 solar terms is published in the Almanac. The date and time for

the four cardinal points in years 2011 through 2020 is shown in Table 4.4. Because

the rough coordination of the date in the Gregorian calendar with vernal equinox, the

dates diﬀer for only one or two days from year to year. Since the ecliptic longitude l of

the Sun at each solar term is well deﬁned, by using the formula of the Sun’s declination

(Eq. 4.27), the declination of the Sun at the 24 solar terms can be calculated,

δ ≈ ε sin l = ε sin



(S − 6) π



, (4.56)

where S is the order of the solar term, shown in Table 4.3. The number 6 is taken

because in Table 4.3, vernal equinox is number 6.

Traditionally, in the Asian calendar, the ﬁrst solar term is deﬁned as spring com-

mences. It is equivalent to deﬁning 3 am early morning as the starting time of a day.

This tradition was established probably because spring commences means the starting

of agricultural activities of a year. The logical starting solar term is the winter solstice,

which is equivalent to deﬁning midnight as the starting time of a day.

4.4 Treatment in Standard Time

For several millennia, all over the world, humans have been using the motion of the Sun

for time keeping, the solar time. However, because the motion of the Sun is not uniform,

solar time shows signiﬁcant deviation from the time deﬁned by uniform motion, for

example, the rotation of Earth, the pendulum, or an atomic clock, represented by the

motion of the ﬁctitious mean sun. The diﬀerence could be as much as 16 min or more.

In order to make a sunlight tracking system based on standard time, the diﬀerence

must be taken into account to reasonable accuracy. In this section, we will present

a treatment which is simple to understand and program and yet accurate enough for

solar energy utilization.

4.4.1 Sidereal Time and Solar Time

To very high accuracy, the angular velocity of Earth’s rotation is a constant. Therefore,

the time interval between two consecutive passages of a given ﬁxed star over an ob-

server’s meridian is a constant, which is an accurate measure of time, the sidereal day.

The word sidereal was derived from the Latin sideus, which means “star”. However,

because Earth also has an orbital motion around the Sun, the duration of the solar day

is diﬀerent from the sidereal day; see Fig. 4.12. For example, at midnight, an observer

sees a distant ﬁxed star at its meridian. At the time the same star passes the meridian

again, Earth has rotated about the Sun by an angle

Δφ =

360

◦

365.2422

=59





. (4.57)

4.4 Treatment in Standard Time  95

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96  Tracking Sunlight

Figure 4.12 Sidereal time and solar time. Because Earth has a rotation on its axis and an

orbital motion around the Sun, the duration of the solar day is diﬀerent from the sidereal day. A solar

day is 0.273% longer than the sidereal day.

Therefore, a solar day is 0.273% longer than the sidereal day. Because solar time is

takenastimeasweknow,andrememberthata360

◦

angle of rotation corresponds to

24 h, it equals

mean solar time =

× 366.2422

365.2422

=24

sidereal time. (4.58)

the length of one sidereal day is

sidereal time =

× 365.2422

366.2422

=23

mean solar time. (4.59)

In the above equations, we introduced the term mean solar time. Hereweareusing

a simpliﬁed model for the orbital motion of Earth around the Sun, namely, a perfect

circle on the sidereal equator with a uniform angular speed. From the point of view

of an observer on Earth, such a ﬁctitious Sun of uniform motion along the celestial

equator is called the mean sun.

From Fig. 4.12, the actual value of time depends on the location of the observer.

Speciﬁcally, it depends on the longitude of the observer. To make a universal time, a

standard longitude must be selected. Similar to the case of the origin of longitude, the

standard longitude for the universal time is selected as the prime longitude located at

Greenwich. The time starting at midnight at Greenwich is called Greenwich mean time

(GMT), or more often, universal time (UT).

The world is divided into time zones. Each zone has a deﬁnition of time which

diﬀers mostly an integer number of hours from the GMT or UT. For example, Eastern

standard time (EST) is deﬁned as UT−5h. In the summer, Eastern daylight saving

time (EDT) is deﬁned as UT−4h.

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4.4.2 Right Ascension of the Sun

As discussed in section 4.2, in the equatorial coordinate system, the position of a star

can be characterized by the declination and the hour angle. The declination of a star

does not vary over time. However, because of the rotation of Earth, the hour angle of

a star varies over time. Uusing the right ascension α instead of the hour angle, the

coordinates of each ﬁxed star are, to a high degree of accuracy, ﬁxed.

Similar to the case of longitude on Earth, a ﬁxed point on the celestial sphere should

be chosen as the reference point. The point universally chosen is the vernal equinox Υ,

the point where the Sun passes the celestial equator while heading north; see Fig. 4.13.

The angle between the intersection of the meridian passing through the star S with

the equator B and the vernal equinox Υ is called the right ascension of the star. The

sign convention is opposite to that of the hour angle: It is measured eastward. For

the treatment of the motion of the Sun, this convention is natural, because it increases

with the number of the day in the year;

4.4.3 Time Diﬀerence Originated from Obliquity

As mentioned previously, there are two origins of the diﬀerence between the motion of

the true sun and the mean sun. The ﬁrst is the obliquity ε. Time is measured by the

angle of the mean sun on the equator, but the true sun is moving on the ecliptic plane.

Assuming that the Sun is moving uniformly on the ecliptic circle, its mean longitude l

is a linear function of time. The mean longitude of the Sun increases by 2π during a

calendar year; see Fig. 4.13. Taking the number of the day N as the unit of time and

the vernal equinox Υ as the origin, which is March 20 or 21, approximately the 80st

day of the year, the mean longitude in radians is

l =

2π (N − 80)

365.2422

. (4.60)

Figure 4.13 Obliquity and

equation of time. Because of

obliquity, even if Earth is orbit-

ing the Sun with a uniform an-

gular speed, the projection of the

speed of the apparent motion of

Sun on the equator is not uni-

form. It gives rise to a term of

the diﬀerence between mean so-

lar time and true solar time with

a periodicity of half a year.

4.4 Treatment in Standard Time  97

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98  Tracking Sunlight

The number of the day N can be computed using Eq. 4.28. The accurate date and

time for vernal equinox is shown in Table 4.4.

Because of obliquity, the projection of the longitude l on the celestial equator, the

right ascension α, is not linear over time. Using the formula for a rectangular spherical

triangle, we ﬁnd the relation

tan α =cosε tan l. (4.61)

Currently, ε ≈ 23.44

◦

, and cos ε ≈ 0.917, very close to 1. Using a trigonometry identity

cos ε =

1 − tan

1 + tan

, (4.62)

Eq. 4.61 can be written as

tan l − tan α

tan l +tanα

=tan

. (4.63)

Using the obvious relation

tan l − tan α

tan l +tanα

sin l cos α −cos l sin α

sin l cos α +cosl sin α

sin(l −α)

sin(l + α)

, (4.64)

we ﬁnd

sin(l −α)

sin(l + α)

=tan

. (4.65)

Because l −α is a small quantity, and hence sin(l −α) ≈ l −α and sin(l + α) ≈ sin(2l),

we obtain a formula up to the ﬁrst order of ε

l −α ≈ tan

sin(2l) ≈ 0.043 sin(2l). (4.66)

4.4.4 Aphelion and Perihelion

The second diﬀerence between the ﬁctitious mean sun and the true sun is that the orbit

of Earth is elliptical rather than circular. The distance between Earth and the Sun is

farthest at aphelion and closest at perihelion. Aphelion is derived from the Greek words

apo (“away from”) and helios (“sun”), while perihelion includes the Greek word peri

(“near”). The date and the hour in the day of each aphelion and perihelion for 2011

to 2020 are shown in Table 4.4. (The accuracy of the time for aphelion and perihelion

is only up to the hour, whileas the accuracy of equinoxes and solstices is up to the

minute). The distance of the Earth and the Sun is about 3% further at aphelion than

it is at perihelion. Because radiation intensity is inversely proportional to the square of

distance, the solar radiation power in early January is 6% stronger than that in early

July.

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Table 4.4: Cardinal Points in Years 2011 through 2020

Year Perihelion Aphelion Equinoxes Solstices

2011 Jan 3, 19h Jul 4, 15h Mar 20, 23:21 Jun 21, 17:16

Sep 23, 09:05 Dec 22, 05:30

2012 Jan 5, 01h Jul 5, 04h Mar 20, 05:14 Jun 20, 23:09

Sep 22, 14:49 Dec 21, 11:12

2013 Jan 2, 05h Jul 5, 15h Mar 20, 11:02 Jun 21, 17:16

Sep 22, 20:44 Dec 22, 05:30

2014 Jan 4, 12h Jul 4, 00h Mar 20, 16:57 Jun 21, 16:38

Sep 23, 02:29 Dec 22, 04:48

2015 Jan 4, 07h Jul 4, 15h Mar 20, 22:45 Jun 21, 17:16

Sep 23, 08:21 Dec 22, 05:30

2016 Jan 2, 23h Jul 4, 16h Mar 20, 04:30 Jun 20, 22:34

Sep 22, 14:21 Dec 21, 10:44

2017 Jan 4, 14h Jul 3, 20h Mar 20, 10:29 Jun 21, 04:24

Sep 22, 20:02 Dec 21, 16:28

2018 Jan 3, 06h Jul 6, 17h Mar 20, 16:15 Jun 21, 10:07

Sep 23, 01:54 Dec 21, 22:23

2019 Jan 3, 05h Jul 4, 22h Mar 20, 21:58 Jun 21, 15:54

Sep 23, 07:50 Dec 22, 04:19

2020 Jan 5, 08h Jul 4, 12h Mar 20, 03:50 Jun 20, 21:44

Sep 22, 13:31 Dec 21, 10:02

4.4.5 Time Diﬀerence Originated from Eccentricity

The eccentricity of the orbit of Earth around the Sun gives rise to a second term in the

equation of time. According to Kepler’s ﬁrst law, the orbit of Earth around the Sun is

an ellipse, and the position of the Sun is at a focus of the ellipse. From the point of

view of Earth, the Sun is orbiting Earth along an ellipse,

r =

1+e cos(θ − θ

)

, (4.67)

where r is the instantaneous distance between the Sun and Earth; e is the eccentricity

of the ellipse, currently e =0.0167; θ is the true longitude of the Sun along the ecliptic;

is the true longitude of the perihelion; and p is a constant.

According to Kepler’s second law, the radius vector of the ellipse sweeps out equal

4.4 Treatment in Standard Time  99

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100  Tracking Sunlight

Figure 4.14 Eccentricity of Earth’s

orbit: Kepler’s laws. According to

Kepler’s laws, Earth is moving around

the Sun on an elliptical orbit. It gives

rise to a term of the diﬀerence between

mean solar time and true solar time with

periodicity of a year.

areas in equal times:



dθ =



[1 + e cos(θ − θ

)]

dθ ∝ t. (4.68)

Because the eccentricity e is small, the integrand of Eq. 4.68 can be expanded into a

power series,

t ∝



[1 − 2e cos(θ − θ

)] dθ = θ − 2e sin(θ − θ

). (4.69)

For an entire year, the longitude increases by 2π, and the sine function returns to its

original value.

Now, we relate the true longitude of the Sun, θ, with the mean longitude l discussed

in the last section, which is proportional to time. For a calendar year, both quantities

increase by 2π. Therefore,

l = θ − 2e sin(θ − θ

). (4.70)

If the eccentricity e is negligibly small, we have

θ = l. (4.71)

For a ﬁrst-order approximation, we substitute θ in the second term of Eq. 4.70 with l,

which yields

θ = l +2e sin(l − l

), (4.72)

where l

is the longitude of perihelion. The second term in Eq. 4.72 arises from the

eccentricity of the orbit.

4.4.6 Equation of Time

By combining Eq. 4.72 with Eq. 4.66, we obtain a complete expression of the equation

of time (ET) up to the ﬁrst order of obliquity and the ﬁrst order of eccentricity, e =

0.0167,

ET = l − α =tan

sin(2l) −2e sin(l − l

)

≈ 0.043 sin(2l) − 0.0334 sin(l − l

(4.73)

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The unit of angles, l and α, is in radians. A complete circle is 2π. To convert the

unit to time as we know, we notice that the mean Sun revolves around Earth every

24 hours. The time it takes for the mean Sun to catch up the true Sun, or vice versa,

following Eq. 4.73, is

ET(min) =

24 × 60

2 π

[0.043 sin(2l) − 0.0334 sin(l − l

)]

=9.85 sin(2l) − 7.65 sin(l − l

(4.74)

In Eq. 4.74, as usual, the time diﬀerence is expressed in minutes. Using the approximate

formula for the mean longitude l, Eq. 4.27, an explicit expression of the equation of

time can be obtained,

ET =



9.85 sin



4π (N − 80)

365.2422



− 7.65 sin



2π (N − 3)

365.2422



(min). (4.75)

Here the date of perihelion is assumed to be January 3. The number of the day in a

year N can be computed using Eq. 4.28. Equation 4.75 is suﬃciently accurate to deal

with problems in sunlight tracking. A chart is shown in Fig. 4.15. Remember that

hour angle ω is to the negative of right ascension α, the equation of time ET should be

added to the hour angle of the Sun, which means that if ET is positive, the real Sun is

faster than the mean Sun.

Figure 4.15 Equation of time. Thick solid curve, the diﬀerence between mean solar time and

true solar time, the so-called equation of time has two terms. The ﬁrst term, the thin solid curve, has

a periodicity of a year, originated from the eccentricity of the orbit, which starts at aphelion. The

second term, the dashed curve, originated from the obliquity of the ecliptic, has a periodicity of half

a year, and starts at the vernal equinox.

4.4 Treatment in Standard Time  101