B2 a21 e2

FlG. 5.15. The celestial sphere and the plane of the ecliptic.

where year is years AD. The arc SD = 6 is the solar declination with a maximum value equal to e. The arc YD = a is called the right ascension of the Sun, and corresponds to the solar longitude as measured on the celestial equator, measured from the vernal equinox where a = 0. Thus, a and 6 define the coordinates of the Sun, point S, on the celestial sphere. The arc A is the ecliptic longitude that is zero at the vernal equinox and measured in the ascending mode. Thus the vernal equinox, or first point of Aries, Y, is the point of reference for the Sun's apparent motion.

5.6.3 Sun-Earth distance and solar longitude The equation for the Sun-Earth distance is given by a(1 - e2)

where v is arc AS, and is measured from perigee and is called the true anomaly, given by v = A - u, (5.41)

and u is the solar longitude of perigee that is equal to w + n, where w is the Earth's longitude of perihelion, the direction of perihelion (as viewed from the Sun) is opposite to the direction of perigee (as viewed from the Earth). Thus, v = 0 at perigee or perihelion, on about 3 January, and v = n at apogee or aphelion, about 3 July. Thus, rp = a(1 — e) at perihelion, shortest Earth-Sun distance, and ra = a(1 + e) at aphelion, largest Sun-Earth distance, and the mean Sun-Earth distance r = 0.5(rp + ra) = a. In Table 5.9 are given the values

Table 5.9 The solar longitude and the seasonal cycle.

Y

A=

G

vernal equinox

20 March

G

A

= n/2

summer solstice

21 June

B

A

=W

aphelion

3 July

Y'

A

= n

autumnal equinox

22 September

F

A

= 3n/2

winter solstice

21 December

A

A

= n + w

perihelion

3 January

of A for the various points along the Earth's orbit corresponding to times of the seasonal cycle.

5.6.4 The equation of time

The Sun's apparent elliptical motion around the Earth is not uniform, however, according to Kepler's Second Law, the radius vector SP sweeps out equal areas in equal times. An element of area corresponding to an element of angle, 50 swept out on an ellipse can be written as r(r + 5r)50/2. Neglecting the 5r50 term we have dArea 1 2 d0 .

did 2 dtd

Time, td in days from 1st January as measured by a clock, is based on a fictitious mean Sun moving on the celestial equator at a uniform rate, i.e. circular motion, which takes one year to complete its orbit. Thus, the mean Sun sweeps out an angle of 2n over one clock year. We can thus define a mean longitude, L, of the mean Sun in its apparent motion around the Earth on the celestial equator, and neglect the time dependence of r, hence, we can write in terms of the mean solar longitude L

dtd 2 dtd so that the mean Sun sweeps out an area na2 in a clock year. Thus dL

and so constant (5.44)

where P is the period of the Earth's orbit, 365.256 days, and Lo = 1.5581n (Astronomical Almanac 1992) at td = 0 midnight on the 1st January. Thus, as can be seen in Fig. 5.15, the mean Sun would be at some right ascension given by a + arc DD', instead of at its true right ascension of a. For the rotating Earth at the centre of the celestial sphere, an observer located on the celestial meridian PGE, sees the Sun at an hour angle GPS, where the hour angle of the observer is defined by h = (t — 12)n/12, (5.46)

where now t is the clock time in hours, so that h is equal to —n radians at midnight when t = 0, zero at noon t = 12, and n radians at midnight again. Thus, the mean sun is located at an hour angle of GPS' instead of at the true hour angle GPS. Thus, we can define an equation of time, E, which is the difference between the true hour angle, h, of the Sun and the hour angle, hm, of the mean Sun

Now, every 15° of hour angle or of mean longitude corresponds to one hour of clock time, so that the correction that needs to be applied to the clock time to obtain the sundial time is E/15 h, with E in degrees. The true hour angle is then given by h(t) = (t — 12 + E/15)n/12. (5.48)

The equation of time varies from about -14 min in February to +16 min in November (Norton's Star Atlas). If a reference observer is at the Greenwich meridian (longitude $ = 0) then the clock time t is called Universal Time (UT) beginning at midnight, and so for another observer located at longitude $ in degrees, the local hour angle is given by h(t) = (t — 12 + $/15 + E/15)n/12. (5.49)

The mean Sun anomaly, g, is different to the true Sun anomaly v, and is given by g = L — u, (5.50)

through a small correction according to v = g + 2e sing + 5e2 sin2g/4 (5.51)

keeping second-order terms in the eccentricity e. The mean anomaly can be calculated from g = 1.986n + 2ntd/P, (5.52)

where td = n — 1+t/24, n is the day of the year with n = 1 on January 1, and t is UT in hours. The above expressions give the true anomaly to a precision of 0.01° between 1950 and 2050. We can now calculate the mean anomaly and hence the true anomaly and calculate the Sun-Earth distance r. Thus, at perigee, when td = 2.625, v = g = 0, and the solar longitude is 283.05° and w° = 103.05° = 0.5725n. The Earth's longitude of perihelion varies from 0 to 360° in about 21 000 years so that w = wo + 9.44 x 10~5n(year — 2000), (5.53)

to a good approximation.

SOLAR FLUX AT THE EARTH'S ORBIT Table 5.10 Orbital variation of incoming solar flux.

Day

Date

Time of season

S(t)/SQ

3

3 January

perihelion

1.03426

80

20 March

vernal equinox

1.00794

173

21 June

summer solstice

0.96820

185

3 July

aphelion

0.96742

266

22 September

autumnal equinox

0.99297

356

21 December

winter solstice

1.03339

5.6.5 Incoming radiation at TOA

We shall now derive the equations for computing the incoming solar radiation at the top of the atmosphere (TOA), for any day of the year and latitudinal location.

The solar flux reaching the Earth's orbit varies with the time of year according to

this variation is given for times of the seasonal cycle for 2000 AD in Table 5.10.

The incoming solar flux, FQ, above the Earth's atmosphere, at a particular location on Earth (given by the longitude, and the latitude, 0) and at a particular time (given by the time of day, t in UT, the day of the year, n, and the year), is given by

with p = cos z, z is the solar zenith angle at any time t during the day and can be calculated from p(t) = A + B cos h(t), (5.56)

with

where 5 and h are the Sun's declination and hour angle, in radians, respectively. The solar declination is given by

and we note that at the equinoxes it is zero. The true solar longitude can be calculated from

To calculate the true hour angle h we need to calculate the equation of time E, eqn (5.47), from L and the solar right ascension a, which can be calculated from a = A - u sin 2A + (u2/2) sin4A, (5.61)

We can also calculate the day length or duration of daytime from the times of sunrise and sunset by setting j = 0. This gives the corresponding hour angles —H and H for sunrise and sunset, where

The day length, as a fraction of one day, is then d = 24H/n. Furthermore, we can calculate the mean value ¡jn between sunrise and sunset on day n and obtain

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