Chapter 3 Energy Balance and Temperature Astro 9601 1
Topics to be covered Energy Balance and Temperature (3.1) - All Conduction (3..1), Radiation (3.. and 3...1) Convection (3..3), Hydrostatic Equilibrium (3..3.1), First Law of Thermodynamics (3..3.) and Adiabatic Lapse rate (3..3.3) All to be discussed in lecture notes with Ch. 4 (where it makes sense!)
Radiation and Planetary Science All solar system bodies are illuminated by the sun Balance between solar radiation received (plus any internal energy) and that emitted defines temperature ultimately equilibrium is reached which defines T Temperature of bodies critical to behaviour of atmospheres, surfaces and interiors 3
Energy Transport Energy can be transmitted by: 1. Conduction. Radiation 3. Convection One mechanism usually dominates In solids, conduction dominates In space and tenuous gases, radiation dominates Convection is important in atmospheres (and liquid interiors) 4
Temperature The temperature of an object is proportional to the average translational kinetic energy of its molecules. Note that one object can have many temperatures 5
Blackbody - Introduction Blackbody a hypothetical (idealized) body that Absorbs all incident radiation (hence the term black ) Emits the maximum possible radiant energy in all wavelength bands in all directions No radiation is reflected All bodies with temperatures above absolute zero emit radiation Max Planck http://home.wanadoo.nl/paulschils/07.0.html 6
The amount of radiation emitted by a blackbody is uniquely determined by its temperature (Planck s law): The black body specific intensity or brightness is defined (following discovery by Max Planck in 1900) as either ( T ) = hc 5 λ 1 B λ ( hc / λkt e 1 or where c=.99x10 10 cm/s, h=6.57x0-7 erg s, k=1.38x10-16 erg/s. Using cgs units (λ in Angstroms) we have B 3 ( = h ν T ) c e 1 ν ( hν / kt 1 B λ ( T ) = 1.1919x 10 e 1.44 x10 8 7 λ 5 / λt λ 1 Max Planck http://home.wanadoo.nl/paulschils/07.0.html 7
Blackbody radiation is isotropic; the radiance is independent of direction Units are J m - Hz -1 s -1 ster -1 (erg cm - Hz -1 s -1 ster -1 ) Recall 10 7 ergs = 1 J B ν 3 hν 1 ( T ) hν / c e = kt 11 http://www.tpub.com/content/neets/1418/css/1418_179.htm 8
Characteristic shape for blackbody radiation plotted using Planck s law Sharp short wavelength cutoff, steep rise to the maximum, gentle dropoff toward longer wavelengths often can use limiting expressions at high f (Wien Law) or low f (Rayleigh-Jeans Law) 9
Classical Limit (small f, large λ) In the limit of small f: B ( T ) B v ν k c B T Rayleigh-Jeans 1 4 λ This equation doesn t involve Planck s constant was originally derived from purely classical considerations. Classical physics predicts the so-called ultraviolet catastrophe an infinite amount of energy being radiated at high frequencies or short wavelengths (derived from the equipartition theorem). 10
At the other extreme for high f (or for short wavelengths), Planck s law simplifies to Wiens Law: 3 hν Bv ( T ) e c B λ hc hν k T 5 hc λ exp λk T B Max Planck http://home.wanadoo.nl/paulschils/07.0.html 11
The Wien displacement law Using Planck s law and differentiating to find the peak (ie. solve B/ λ=0), one can find the wavelength of peak emission for a blackbody at temperature t T: λ m = 897 ( μm K) T known as the Wien displacement law. This law makes possible the estimate of the temperature of a radiation source from knowledge of its emission spectrum. 1
The Wien displacement law Consequence: solar radiation (due to the temperature of the sun) is concentrated in the visible and near-ir parts of the spectrum planetary radiation and that of their atmospheres is largely confined to the IR (normalized) 13
The Wien displacement law Note the lack of overlap that allows separation of the radiative transfer problems of the earth and of the sun 14
The Stefan-Boltzmann law If we integrate Planck s law just above the surface of an object and over all frequencies, we find: 0 F ( T ) F dν = π B ( T ) dν ν F = σ T where F is the flux (power/unit area) which h is known as the Stefan-Boltzmann law 0 4 ν F = Flux, (power/unit area), T = Temp. in Kelvin, σ = 5.67 x 10-8 W/m K 4 (conductivity) For non-ideal black body, F = σt 4 ε where ε = emissivity < 1. Josef Stefan http://home.wanadoo.nl/paulschils/07.0.html 15
Albedos When the sun illuminates an object, some of the radiation is absorbed, and some scattered. The albedo (ratio of reflected and scattered intensity to incident intensity) varies with wavelength. A ν is the monochromatic albedo. The luminosity observed depends on the geometry, specifically the phase angle. Earth Object Sun 16
Albedos The geometric albedo is F ( ϕ = 0) the ratio of the flux reflected A0 = F incident head-on (back to the sun) to the incident flux The bond albedo is the ratio of the total flux reflected to the incident. It incorporates an integral Ab = A0 q ph over phase angle 17
Marley et al. (1999) 18
Phase Function: φ = I( ϕ) I(0) Sudarsky et al. (005) 19
Eros from NEAR Muinonen et al. (00) 0
Equilibrium temperature The sunlit hemisphere of a planet absorbs radiation: Lsun of planet Fin = ( 1 Ab ) πr 4πr Cross-sectional area Area over which solar radiation is spread at distance r from sun If the planet rotates rapidly, its temperature is uniform. In that case, it emits radiation: F out = 4πR εσt 4 We can calculate the equilibrium temperature by setting the two equal to each other. 1
Equilibrium temperature F in = ( 1 Ab ) R 4πr Lsun 4 π F 4 R T out = π εσ We can calculate the equilibrium temperature by setting the two equal to each other. T eq F = r sun (1 Ab ) 4εσ The temperature t depends d on the distance to the sun, but not on the size of the object. 1/ 4 3
Planetary Temperatures Teq Teff Tsurf Mercury 446 K 446 K 100 75 K Venus 38 38 733 Earth 63 63 88 Moon 77 77 77 Mars 15 Jupiter 113 14 Saturn 83 95 Uranus 60 59 Neptune 48 59 4
Albedos in the solar system Rocky surfaces: 0.1 0. Icy bodies 0. 0.7 Gaseous planets: ~0.3 The Moon: 0.07 Venus: 075 0.75 We can measure the visual albedo by comparing the reflected and emitted radiation. 5
Reflected visible light A v =0.0 A v =0.05 005 IR emission 6
Solar radiation flux falling on an asteroid surface per square meter: Total reflected visible luminosity of the asteroid is given by: Energy not reflected is absorbed and then re-emitted at tir wavelengths: Assume asteroid is at opposition with the d Earth and reflects visible radiation uniformly over its sunlit hemisphere (π steradians). Visible radiation detected at the Earth is then: 7
Thermal radiation is reflected in all directions (slow rotator) so as seen at the Earth the thermal radiation received is: Thus the ratio of visible to thermal radiation is: Therefore if we can simultaneously measure the thermal and visible flux we can directly measure the visible (and hence thermal) albedos. 8
Heat Conduction Conduction is the transport of energy by collisions between particles. Conduction is important in the upper atmosphere, where the mean free path is long and collisions are important. Sunlight heats many surfaces during the day. The energy is transported downwards from the surface. The rate of flow of heat is known at the heat flux, Q. Q depends on the temperature gradient, or and the thermal conductivity K T. K T is a measure of the material s ability to conduct heat. Units of K T : erg egs -1 cm -1 K -1 or J s -1 m -1 K -1 9
Conduction as diffusion The energy that goes into a volume element per unit time is: How much does this heat up the material? Combining this with We get: or where This is known as the diffusion equation Compare to the wave equation: which h has oscillating solutions. The diffusion equation has exponentially spreading solutions. 30
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t t t t t 3
Thermal diffusion coefficients C P (J/kgK) ρ (kg/m (g 3 ) K T (W/mK) K d (m /s) Water 400 1000.18 5.5 x 10-7 Iron 450 7800 80.3 x 10-5 Stone 700 3000-7.3 x 10-5 Typical Near-Earth Asteroid rotation period ~ 10 4 sec Z ~ 10 cm Longest known asteroid rotation period ~ 10 7 sec Z ~ 10 m For Mars/Moon Z ~ 5 cm 33