Lecture_02: Outline Thermal Emission Blackbody radiation, temperature dependence, Stefan’s law, Wien’s law Statistical mechanics, Boltzmann distribution, equipartition law Cavity radiation, Rayleigh-Jeans classical formula

Matter under irradiation: Blackbody radiation Matter under irradiation:

Colors of matter: This is not an emission! Absorbs everything: black Blackbody radiation Colors of matter: Absorbs everything: black Reflects everything: white Transparent for everything: no color Absorbs everything, but yellow: yellow Absorbs everything, but red: red This is not an emission!

Emission of heated iron: Blackbody radiation Emission of heated iron:

Emission of heated iron: Blackbody radiation Emission of heated iron:

Emission of heated iron: Blackbody radiation Emission of heated iron:

Stefan’s Law: Blackbody radiation Radiancy, RT, is the total energy emitted per unit time per unit area from a blackbody with temperature T Spectral radiancy, RT(ν), is the radiancy between frequencies ν and ν+dν RT(λ) Stefan-Boltzmann constant:

Wien’s displacement law: Blackbody radiation Wien’s displacement law:

Blackbody radiation Temperature of stars: Sun: λmax = 5.1ּ10-7m; North Star: λmax = 3.5ּ10-7m

Blackbody radiation Temperature of stars: Sun: λmax = 5.1ּ10-7m; North Star: λmax = 3.5ּ10-7m Sun: T = 5700K; North Star: T = 8300K

Temperature of stars: Radiancy of stars: Blackbody radiation Temperature of stars: Sun: λmax = 5.1ּ10-7m; North Star: λmax = 3.5ּ10-7m Sun: T = 5700K; North Star: T = 8300K Radiancy of stars:

Temperature of stars: Radiancy of stars: Blackbody radiation Temperature of stars: Sun: λmax = 5.1ּ10-7m; North Star: λmax = 3.5ּ10-7m Sun: T = 5700K; North Star: T = 8300K Radiancy of stars: Sun: RT = 5.9ּ107 W/m2; North Star: RT = 2.71ּ108 W/m2

Ideal gas: Statistical mechanics Temperature is related to averaged kinetic energy For each of projections (degrees of freedom):

Theorem of Equipartition of Energy: Statistical mechanics Theorem of Equipartition of Energy: Each degree of freedom contributes ½kT to the energy of a system, where possible degrees of freedom are those associated with translation, rotation and vibration of molecules

Boltzmann Distribution: Statistical mechanics Boltzmann Distribution: Distribution function (number density) nV(E): It is defined so that nV(E) dE is the number of molecules per unit volume with energy between E and E + dE Probability P(E): It is defined so that P(E) dE is the probability to find a particular molecule between E and E + dE

Statistical mechanics Free particle: Harmonic oscillator:

Averaged values: Averaged energy: Averaged velocity: Usually: Statistical mechanics Averaged values: Averaged energy: Averaged velocity: Usually:

Rayleigh-Jeans formula Cavity radiation: A good approximation of a black body is a small hole leading to the inside of a hollow object The hole acts as a perfect absorber Energy density, ρT(ν), is the energy contained in a unit volume of the cavity at temperature T between frequencies ν and ν+dν

Electromagnetic waves in cavity: Rayleigh-Jeans formula Electromagnetic waves in cavity: E = Emax sin(2πx/λ) sin(2πνt) With metallic walls, E = 0 at the wall (x = 0,a)! a

Electromagnetic waves in cavity: Rayleigh-Jeans formula Electromagnetic waves in cavity: E = Emax sin(2πx/λ) sin(2πνt) With metallic walls, E = 0 at the wall (x = 0,a)! a Only standing waves are possible! n = 1 n = 2 2a/λ = n n = 3 2aν/c = n n = 4

Allowed frequencies: ν = cn/2a Polarization Rayleigh-Jeans formula 1 2 3 4 5 n Number of allowed waves, N(ν)dν, between frequencies ν and ν+dν Polarization

Two independent waves for each ν, λ Rayleigh-Jeans formula Polarization: E = Emax sin(2πx/λ) sin(2πνt) Two independent waves for each ν, λ

One-dimensional case: Rayleigh-Jeans formula One-dimensional case: Three-dimensional case: Energy spectrum, ρT(ν)dν, is equal to the number of allowed wave per unit volume times the energy of each wave Each wave can be considered as a harmonic oscillator with

Rayleigh-Jeans formula: