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2.2 Celestial mechanics

Radiation and Climate Change FS 2017 Martin Wild

Earth‘s orbit

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


semi major axis

major axis minor axis

Definitions: Perihelion P: point on the orbit which is closest to the Sun (Jan 3)

Aphelion A: point on the orbit which is farthest from the Sun (July 5)

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.

0 ≤ e ≤1


Earth‘s orbit


Earth‘s orbit

Variation of eccentricity over Earth’s history

Today e=0.0167

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


a semi major axis

S

Earth‘s orbit

Earth-Sun distance varies over the course of a year: ⇒  Insolation Sr at distance r:

Using

a: semi major Axis rs: radius of the sun

S: solar constant at distance a from sun

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


€

Sr (r) =4πrs

2Bs

4πr 2=4πa 2S4πr 2

= Sar

#

$ %

&

' (

2

a

Sr S

Earth‘s orbit

Sr (a) =4πrs

2Bs4πa2

≡ S ⇒ Bs =Sa2

rs2


Earth‘s orbit Deviation from solar constant due to Earth orbit Special cases:

Earth in Aphel:

r=a+ae=a(1+e)

=>


2

2

)1()1())1(()(

eS

eaa

SeaSperihelS rr −=⎟

⎟⎠

⎞⎜⎜⎝

⎛

−=−=

2

2

)1()1())1(()(

eS

eaa

SeaSaphelS rr +=⎟

⎟⎠

⎞⎜⎜⎝

⎛

+=+=

2

)( ⎟⎟⎠

⎞⎜⎜⎝

⎛=

ra

SrSr

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

Earth‘s orbit

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

=>


€

Sr (perihel)Sr (aphel)

=(1+ e)2

(1− e)2=(1+ 0.0167)2

(1− 0.0167)2=1.07

27.1)06.01()06.01(

)1()1(

)()(

2

2

2

2

=−

+=

−

+=

ee

aphelSperihelS

r

r

€

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

Earth‘s orbit


Earth‘s orbit

Deviation from solar constant due to Earth orbit

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

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

Total solar energy per m2 distributed over Earth sphere S R2*π / (4 R2*π) = S/4 ~ 340 Wm-2


2

44)( ⎟

⎟⎠

⎞⎜⎜⎝

⎛==

raSSrG r

Solar radiation reaching planet Earth

Total solar energy received on Earth:

4πr2 G =4π(6.37 106m)2 340*Wm-2 =1.74 1017W (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 TerraW (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



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Solar thermal power

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Planetary albedo A: Fraction of reflected solar radiation with respect to incoming solar radiation

Mean annual energy GA absorbed by the planet per m2 on the sphere:

A = 0.3 for Earth


GA =

S4(1−A)


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

Effective Temperature

Teff =

GAσ

4 = 240Wm−2

5.67*10−8Wm−2K −44 =255K

GA =

S4(1−A)=σTeff

4

Effective Temperature


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

•  If the radiation 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 radiation 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) Teff (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

Exercices

3) How does the effective Temperature Teff, P for a given planet depend on its distance r to the sun, Teff= Teff (r)?


Effective Temperature of Planets


Log

Tem

pera

ture

(K)

Log Distance from sun r

Teff (r)∝1r

Teff ∝1r= r

−12

log Teff( )∝ log r−12

#

$%

&

'(

log Teff( )∝− 12 log r( )y = −0.5x

Habitable Zones


Habitable Zones


Radiative criteria for life on planets (I) Planet with albedo αp and solar constant Sp:

Distance rp to star (with emission Temperature Tstar and radius rstar) so that effective planetary Temperature Tp eff in a range that allows for liquid water on the planet:

Inner limit: Tp eff below 373K:

Outer limit: planet heats up above 273K:

Tp eff =Sp(1−α p )4σ

4 =4πrstar

2 Bstar (1−α p )4πrp

2 4σ4 =

rstar2 σTstar

4 (1−α p )4rp

2σ4 = Tstar

rstar2 (1−α p )4rp

24

⇒min rp =rstarTstar

2 (1−α p )2Tp eff

2 =rstarTstar

2 (1−α p )2*3732

⇒maxrp =

rstarT2star

2Tp eff 2=rstarT

2star 1−αP( )2*2732

Habitable Zones


Radiative criteria for life on planets (II): Not only quantity of radiation important (which determines planetary temperature), but also quality (spectral composition, photon energy)

If Tstar >> T sun (5778 K): > more emission in shorter wavelenghts (UV range) (c.f Wien‘s law)> individual photon highly energetic (more energetic than binding energy of organic molecules)

> destroyes organic molecules

Our sun already too much UV> biosphere in water produced oxygen during Earth history that allowed the build up of ozone layer to shield UV > prerequisite for development of life on land (450-500 Mio years ago)

=> Biosphere itself made our planet habitable

Habitable Zones


Radiative criteria for life on planets (III): Not only quantity of radiation important (which determines planetary temperature), but also quality (spectral composition, photon energy)

If Tstar << T sun (5778 K): less emission in high frequency (UV) ranges > photons have too low energy (E=hv) to allow for photochemistery > photosyntheis would not work

=> Medium sized stars have an ideal Planck curve for life on planets


Insolation at a specific location and time

S(1− A)4

=σTeff4

Zero-dimensional climate model: Only global mean insolation required

Three-dimensional climate model: Insolation required for any given location at any given time

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