The Sun. Solar radiation (Sun Earth-Relationships) The Sun. The Sun. Our Sun

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Solar radiation

(Sun Earth-Relationships)

Radiation and Climate Change FS 2012 Martin Wild

The Sun

Solar Factoids (I)

•  The sun, a medium-size star in the milky way galaxy, consisting of about 300 billion stars.

•  A gaseous sphere of radius about 695‘500 km (about 109 times of Earth radius) => by far the largest object in the solar system

•  Mass: 1.989 * 10³⁰kg (99.8% of total mass of solar system)

Our Sun

The Sun

Solar Factoids (II)

•  Sun consists of 3 parts of hydrogen, one part of helium. Proportion changes over time.

•  Sun‘s energy output is produced in the core of the sun by nuclear reactions (fusion of four hydrogen (H) atoms into one helium (He) atom).

•  Sun is about 4.5 billion years old. Since its birth it has used up about half of the hydrogen in its core.

•  Sufficient fuel remains for the Sun to continue radiating "peacefully" for another 5 billion years (although its luminosity will approximately double over that period), but eventually it will run out of hydrogen fuel.

The Sun

Solar Factoids (III)

•  The Sun's energy output is 3.84 * 10¹⁷Gigawatts: (a typical nuclear power plant produces 1 Gigawatt)

•  The outer 500 km of the sun (“photosphere“) emits most of radiation received on Earth

•  Radiation emitted by the photosphere closely approximates that of a blackbody of 5777K

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The Sun Emission of Sun

Effective surface temperature of the sun: 5778 K

=> Emission B_s (per m²⁾ at the sun surface (Stefan-Boltzman law):

B

_s

= ! T

⁴

=5.67 10

^-8

Wm

^-2

K

^-4

*(5778K)

⁴

= 6.32*10

⁷

Wm

^-2

=> Total emission of Sun E_TOT:

E

_TOT

=4 " r

_s²

B

_s

with r

_s

=6.955 *10

⁸

m= radius of the sun:

4 * 3.14* (6.955 *10⁸m)²*6.32 10⁷Wm^-2=3.84 10²⁶W= 3.84 10¹⁷GigaW

cf. World‘s energy cosumption: 15 TerraW (1.5 10¹³ W)

Area on Sun surface required to cover world‘s energy cosumption: 1.5 10¹³ W / B_s = 1.5 10¹³ W / 6.3 10⁷Wm^-2=2.5 10⁵m²=0.25 km².

=>if we could harvest energy directly on the sun surface, 0.25 km² would be sufficient to cover world‘s energy demands.

Binding energy per nucleon in He core: 1.1*10^-12 J

!  Energy generated by one fusion reaction combining 4 H nuclei into one He core: 4 *1.1 10^-12 J= 4.4 10^-12J

Total energy per second emitted by sun: E_TOT=3.84*10²⁶W (Js^-1)

!  Number of fusion reactions per second = ETOT/ energy generated per fusion reaction= 3.84 10²⁶Js^-1/ 4.4 10^-12J = 0.9*10³⁸s^-1

1 proton mass= 1.67*10^-27kg

=> per fusion reaction 4*1.67*10^-27kg of H is consumed.

Total amount of H consumed in the Sun per second:

= number of fusion reactions * amount of H consumed per reaction = 0.9*10³⁸s^-1*4*1.67e^-27kg= 6 *10¹¹kg= 600 Mio Tons

=> Every second 600 Mio Tons of H are transformed to He

Solar fusion

Total emission E_TOTof Sun:

E

_TOT =

4 " r

_s²

* B

_s

!  Total Emission of Sun (in W) distributed over a sphere (in m²) with radius a, where a= Earth-Sun Distance (semi major axis of Earth’s orbit, 149.6 * 10⁹m), determines the Solar irradiance S per m² at the Top of the Earth’s atmosphere (Solar Constant) at distance a :

a S=1366Wm^-2

Solar radiation

r_s

S = 4 " r

_s²

B

_s

/ (4 " a

²

) = (r

_s

/a)

²

B

_s

=(6.955*10

⁸

m / 149.6*10

⁹

m)

²

6.3210

⁷

Wm

^-2

= 1366 Wm

^-2

Current best estimate from measurements: 1361 Wm^-2

5 Wm^-2 deviation may to difference from ideal black body and measurement uncertainties

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More generally, if a planet is at distance r_p from the sun, then the solar irradiance S_p (in Wm^-2) onto the planet is:

Intensity of solar irradiance decreases with distance according to Inverse square law.

Solar radiation

Examples:

r_s=6.955 *10⁸m B_s=6.32*10⁷Wm^-2

=>c=3.057* 10²⁵W

Solar radiation

Planet Distance from Sun (10⁹ m)

Intensity of solar radiation (Wm^-2)

Venus 108 2620

Earth 149.6 1366

Mars 228 558

Sun Earth relationships

Earth‘s orbit around the Sun:

Earth's orbit is an ellipse and the sun is located in one of its focal points. Definition Ellipse: The sum of the distances from any point on the ellipse to the two focal points is constant (equal 2 x semi major axis a)

=> Sun Earth-distance r varies during the course of the year

Radiation and Climate Change FS 2012 Martin Wild semi major axis

Definitions:

Perihelion P: point on the orbit which is closest to the Sun Aphelion A: point on the orbit which is farthest from the Sun Eccentricity e: Amount by which orbit deviates from a perfect circle, where 0 is perfectly circular, and 1.0 is a parabola. Ratio of the distance between the foci of the ellipse to the length of the major axis of the ellipse.

Solar radiation

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Definitions:

Solar constant S (1361 Wm^-2): Solar irradiance obtained per m² on a plane perpendicular to the sunbeam at distance a (semi major axis) from the Sun.

Distance a (semi major axis) sometimes also called 1 Astronomical Unit (AU)

# 150 Mio km

Solar radiation

a

semi major axis

S

Earth-Sun distance varies over the course of a year:

!  Insolation Sr at distance r:

a: semi major Axis:

r is a function of time of the year: r(t)

Solar radiation

a

S

_r

S

Special cases:

Earth in Aphel: r=a+ae=a(1+e)

=>

Solar radiation

Earth in Perihel: r=a-ae=a(1-e)

Current e=0.0167:

7% difference in insolation between Perihel and Aphel

max. e in Earth history: 0.06:

27% difference in insolation between Perihel and Aphel

Solar radiation

!

S

_r

( perihel)

S

_r

(aphel) ⁼

(1 + e)

²

(1 " e)

²

⁼

(1 + 0.0167)

²

(1 " 0.0167)

²

^{= 1.07}

!

Sr( perihel) Sr(aphel) ⁼

S (1 " e)²

S (1 + e)²

= ^S

(1 " e)² (1 + e)²

S ⁼

(1 + e)² (1 " e)²

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Insolation G(r) received per m² on average on the Earth sphere

Total energy taken out of solar flux by Earth disk: S R²*"

Total solar energy per m² distributed over Earth sphere S R²*" / (4 R²*") = S/4= 340 Wm^-2Radiation and Climate Change FS 2012 Martin Wild

Solar radiation Solar radiation

Mean insolation on Earth G_Pover an entire orbital period P (annual mean insolation)

Integrating:

yields:

where e is the eccentricity of the elliptic orbit of Earth around the sun

!

G

_P

₌ ^S

4P

a

r(t)

"

# $ ^%

&

'

2

dt

0 P

( ⁼ ^S

4 1 ) e

²

Solar radiation

Current conditions: e=0.0167:

=>e effect:0.00014*340Wm^-2=0. 033 Wm^-2

Maximum eccentricity over past Million years: e=0.06:

=>e effect:0.0018*340Wm^-2=0. 61 Wm^-2

! term negligible for annual mean calculations

=> G_P=S/4= 340Wm^-2annual mean insolation

Total solar energy received on earth:

4"r

²

G

_p

=4"(6.37 10

⁶

m)

²

340*Wm

^-2

=1.74 10

¹⁷

W (174 PetaW)

(174,000,000,000,000,000 J per second from the sun)

Compare: 1 average swiss nuclear power plant generates power on the order of 1 GigaW=10⁹ W

!  Solar energy incident on Earth compares to about 1.7 x 10⁸ nuclear power plants (170 Mio. nuclear power plants)

Compare: World‘s current energy consumption: 15 TeraW (1.5$10¹³ W)

!  10’000 times smaller than solar energy incident on the planet

!  solar energy received within less than one hour would be sufficient to cover one year of World‘s current energy consumption

Solar radiation

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Desertec: Solar Power from the Desert

www.desertec.org

Within 6h deserts receive more energy from the sun than humankind consumes within a year

Solar radiation

Planetary albedo A:

Fraction of reflected solar radiation with respect to incoming solar radiation

Mean annual energy G_Aabsorbed by the planet per m² on the sphere:

A = 0.3 for Earth

In equilibrium, absorbed shortwave energy G_A (over the Earth disk) is balanced by longwave emission (over the Earth sphere) according to the Stefan-Boltzman law with an effective temperature T_eff:

Effective Temperature Effective Temperature

Effective temperature: (blackbody) temperature at which the emitted longwave equals the absorbed shortwave radiation.

• If the temperature of a planet is below the effective temperature it will emit less radiation than it absorbs => planet will warm until it reaches radiative equilibrium and effective temperature.

• if its temperature is above the effective temperature it will cool toward radiative equilibrium by emitting more radiation than it absorbs.

planet distance from sun (10⁹m)

albedo (1-albedo) T_eff (K)

Mercury 58 0.06 0.94 442

Venus 108 0.78 0.22 227

Earth 150 0.30 0.70 255

Mars 228 0.17 0.83 216

Jupiter 778 0.45 0.55 105

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Effective Temperature of Planets

Temperature (K)

Distance from sun

Exercices

Daily course of sun

Surface insolation I (irradiance) at a specific location and time:

% solar zenith angle at that position and time

Solar Zenith angle % function of:

•  Time of the day, expressed in hour angle H

•  Latitude &

•  Calender day (season), expressed as declination '

r function of time on Earth orbit.

Daily course of sun

Earth centered Cartesian coordinate system (x; y; z)

z-axis points to the North pole

x-axis in the equatorial plane with sun in the x-z-plane

points to local zenith at P points to the Sun

% is zenith angle at observer point P

Determination of zenith angle !

!

n !

!

s !

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Daily course of sun

Declination ": angle between the direction to the sun and to equatorial plane

Determination of zenith angle !

Declination varies over the year from +23°27’ (21. June) to -23°27’ (21. Dec)

Daily course of sun

Hour Angle H:

Angle in the equatorial plane between the meridian of the observer P and the direction to the sun projected onto the equatorial plane.

Hour angle in radiance 0: solar noon

Determination of zenith angle !

Daily course of sun

Determination of zenith angle !

Local Zenith angle: Zenith angle at point P: Angle between local zenith and the direction to the sun

!

n !

!

s !

Daily course of sun

Determination of zenit angle !

Unit vector

1

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Daily course of sun

Determination of zenit angle !

Unit vector

sin '

1

Daily course of sun

Determination of zenit angle !

Unit vector

1

Daily course of sun

Determination of zenit angle !

1

cos"cosH

H

Unit vector

Daily course of sun

Determination of zenit angle !

Unit vector

"#

sin"#

1

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Daily course of sun

direction to the local zenith: direction to the sun at the location of an observer

With the scalar product we obtain the zenith angle % :

Fundamental equation for zenith angle %

The astronomical sunrise and sunset, +-H

₀

, are given for

the mathematical horizon at %="/2

where H

₀

is defined only for -1 ( cosH

₀

( 1. For cos H

₀

>

1, we have the polar night with no sunrise and for cosH

₀

<-1 we have the polar day with no sunset.

Daily course of sun

!

cos" = cos H cos# cos$ + sin # sin $

with " = ^%

2 ^{when H = H}

⁰

0 = cos H

₀

cos # cos$ + sin # sin$

cos H

₀

_{= &} ^{sin # sin$}

cos # cos$ = & tan # tan$

Daily insolation I_d at a given location and date is obtained by integrating

for the hour angle from sunrise at -t₀ to sunset at t₀. Declination is kept constant during one day. Horizon at a zenith angle of 90°, => integral is evaluated from sunrise at –t₀to sunset at t₀, with

where the hour angle H₀ is measured in radian,

The integral can be evaluated analytically.

Daily insolation

The integral can be evaluated analytically,

where H₀is the hour angle for sunrise at the mathematical horizon.

Daily insolation

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!

cos"

# Ho Ho

$

^{dH =}

(cos H cos % cos& + sin % sin&)dH =

# Ho Ho

$

= sin H cos % cos& + sin % sin& ' H _]

_{# H}^H⁰₀

= sin H

₀

cos % cos& + sin % sin& ' H

0

^{# (sin(#H}

0

)cos % cos& + sin % sin& ' (#H

0

⁾⁾

= sin H

₀

cos % cos& + sin % sin& ' H

₀

# (#sin(H

₀

)cos % cos& # sin % sin & ' H

₀

)

= 2(sin H

0

cos % cos& + sin % sin& ' H

0

⁾

( I

d

= S ⁸⁶⁴⁰⁰

2)

a r

*

+ , ^-

. /

2

cos"

# Ho Ho

$

^dH

= 2S ⁸⁶⁴⁰⁰

2)

a r

*

+ , ^-

. /

2

(sin H

₀

cos % cos& + sin % sin& ' H

₀

)

Daily insolation

Mean daily insolation at TOA (in Wm

^-2

The Sun. Solar radiation (Sun Earth-Relationships) The Sun. The Sun. Our Sun

Solar radiation

(Sun Earth-Relationships)

The Sun

Our Sun

The Sun

The Sun

The Sun Emission of Sun

B

= ! T

=5.67 10

Wm

K

*(5778K)

= 6.32*10

Wm

E

=4 " r

B

with r

=6.955 *10

m= radius of the sun:

=> Every second 600 Mio Tons of H are transformed to He

Solar fusion

E

4 " r

* B

Solar radiation

S = 4 " r

B

/ (4 " a

) = (r

/a)

B

=(6.955*10

m / 149.6*10

m)

*6.32*10

Wm

= 1366 Wm

Solar radiation

Solar radiation

Sun Earth relationships

Solar radiation

Solar radiation

a

S

Solar radiation

a

S

S

Special cases:

Solar radiation

Solar radiation

!

S

( perihel)

S

(aphel) =

(1 + e)

(1 " e)

=

(1 + 0.0167)

(1 " 0.0167)

= 1.07

Solar radiation Solar radiation

!

G

= S

4P

a

r(t)

"

# $ %

&

'

dt

( = S

4 1 ) e

Solar radiation

6.3210

(aphel) ⁼

⁼

^{= 1.07}

₌ ^S

# $ ^%

( ⁼ ^S

with " = ^%