Computing Solar Output and Solar Corona Temperature

1
Computing Solar Output and Solar Corona
Temperature 1. Observed Spectrum at Top of
Atmosphere A. Radiation Laws Wiens
Displacement Law Estimate Solar Corona
Temperature Stefan-Boltzmann Law Determine
Solar Flux Density from T 2. Conservation of
Energy Observed Solar Constant Geometrical
Calculation of Solar Flux Density
2
Observed Solar Spectrum
Peak Value at 0.45µ
3
Wiens Displacement Law
T A/? T 2898 µK 6400K .45 µ
Stefan-Boltzmann-Law
F sT4 F (5.67x10-8Wm-2K-4)(6400K)4 F
9.6x107Wm-2 For T 6000K, F 7.3x107Wm-2 For
T 5800K, F 6.4x107Wm-2
4
Conservation of Energy
Earths Observed Solar Constant S 1360Wm-2
The energy passing a sphere at the mean earth-sun
distance, dm is S(4p dm2)
The energy leaving the sun is the flux density
(in Wm-2) multiplied by the surface area of the
sun (4p rs2), where rs is the solar radius.
S(4p dm2) F (4p rs2) or F S (dm/rs)2
Using dm 1.5x108 km and rs 7x105 km, F
6.2x107Wm-2 So, T 5800K ?
5
Comparison of Blackbody Curve with Solar Curve
6
Effective Temperature
Radiative equilibrium temperature of a planet
Using the Stefan-Boltzmann law,a planet emits
over its entire surface (sTe4 )4pR2 watts of
power. R is the planets radius. For
equilibrium, this must be balanced by incoming
solar radiation, S(1-a) pR2
The effective temperature Te S(1-a) /4s1/4
For Earth, S 1360 Wm-2 and a 0.3, Te 255K
http//itg1.meteor.wisc.edu/wxwise/museum/a5/a5run
1.html
7
Basic Sun-Earth Geometry

• Solar Declination Angle
• Solar Zenith Angle
• Solar Hour Angle
• Solar Azimuth Angle

8
Solar Declination AngleThe angle between the
Earths axis and the suns orbital plane (plane
of the ecliptic)
Declination angle d 23.5º at the summer
solstice -23.5 º at the winter solstice d
-23.5 º cos(360/365)(DN 9) where DN is the
number of the day of the year starting with
January 1. The declination angle is equivalent
to the latitude at which the suns direct rays
are vertical.
9
Solar Zenith Angle The angle between local
vertical and the direct solar beam
cos? sinf sind cosf cosd cost where ?
solar zenith angle f local latitude (- for
Southern Hemisphere) d solar declination
angle t solar hour angle
10
Solar Hour AngleLongitudinal position of sun
relative to local longitude
t 0 º at local noon, t 180 º at local
midnight
Computing Solar Hour Angle .From Z Time ? ? (Z
12)15? ? F / 4
Z is Z time ? is local longitude F
is number of minutes the sun is fast Ex
1515 Z15.25 W. longitude is neg. F is
negative for slow sun E. longitude is
pos. From Local Time ? ? (T 12)15?
(? - ??) F / 4 -
D T is local time ? is local longitude D
15? if on daylight Ex 315pm, T15.25 ?? is
base longitude D 0? if on standard time Base
longitude is computed based on difference between
local standard time and Z time. The base
longitude for Denver is -105?, the base longitude
for New York is -75?,etc.
11
Solar Azimuth Angle
A 180 º sin-1 (cos d sin t /sin ?)
12
Denver Sunrise and Sunset Information  
Date Sunrise Sunset Daylength Noon 7Dec 707 16
35 928 1151 22Dec 718 1639
921 1158 5Jan 721 1649
928 1205 31Jan 708 1717 1009 1213   Exa
mple 1 Computing Solar Zenith Angle for 29 JAN
in Denver   Local Latitude 39.75? Declination
Angle -23.5? cos(38)360/365 -18.6?
(-18? from chart)   At noon, Cos(?)
sin(39.75?)sin(-18?) cos(39.75?)cos(-18?)cos(0?)
? 57.8?   At 1 hour before local noon
(1113), ? 15?, so ? 59.4?  
13
Example 2 Find the zenith angle for Buenos Aires
(61?W, 35?S) at 2pm on Feb.1. Buenos Aires is
normally 4 hours behind Z time, but currently on
daylight savings time.   1.     Estimate
declination angle from analemma ? -17? 2.    
Find ? ? ? (T 12)15? (? -
??) F / 4 - D (14
12) 15? (-61 - 60)?
(-14 / 4) ? - 15? (30 -1-3.5-15) ? ? ?
10.5? 3.     Find ? cos ? sin?sin?
cos?cos?cos? cos ? (-.57)(-.29)
(.82)(.96)(.98) cos ? .168 .734 .902 ?
25.6?