Addendum 1: Radiation Laws and The Bohr Atom

First define some constants and dimensional units needed below

1. Wave equation. The wave equation relates a wave's frequency (f), wavelength (l), and speed. For electromagnetic radiation (light, radio waves, x-rays, etc), the speed is the speed of light c. The SI frequency unit is Hertz (abbreviated Hz), which means cycles per second.

Example: A radio wave (FM station KRUI) has a frequency f = 89.7 MHz (1 MHz = 10⁶ Hz). What is the wavelength of this radiation?

2. Doppler equation. If a source of radiation is moving toward or away from an observer, the received frequency (and wavelength) is shifted by amount proportional to the speed of the emitter. If the object is moving away, the received frequency is lower than the emitted frequency. Likewise, the received wavelength is larger for an object moving away.

or

V>0 if source is moving away

Example: A star is observed whose spectral line Ha has a wavelength l = 657.2 nm (1 nm = 10^-9 m). The Ha line is normally l = 656.3 nm in the laboratory. In which direction is the star moving, and at which speed?

3. Energy of a photon. A single photon (quantum of light) has a energy directly proportional to its frequency For many problems, the energy is most conveniently expressed in terms of electron volts (eV). where 1 eV = 1.6x10^-19 Joule.

or

where h is called Planck's constant

Example: What is the energy of a photon with a wavelength of the Balmer 3->2 (Ha, l=656.3nm) transition level in hydrogen?

4. Wien's Law. Wien's law (pronounced Veen's law, German w as v) relates the temperature and wavelength of maximum intensity for any thermal ('black body) emitter.

Note that higher temperature implies shorter wavelength.

Example: A star is observed with a spectrum as shown in the figure. Calculate (approximately) the surface temperature of the star in degrees Kelvin.

5. The Rydberg formula. The Rydberg formula was experimentally determined by Janne Rydberg about 1890, to compute the wavelengths of the lines emitted by hydrogen gas. It was later derived theoretically by Neils Bohr using on his ('Bohr atom') model which assumed quantization of the angular momentum of the electron orbits.

Where the constant RH is the Rydberg constant:

Example : What is the expected wavelength for the 4 -> 2 (Hb) transition for Hydrogen?

Note: the plot above is given is Angstrom units (1 A = 0.1 nm)

5'. Derivation of Rydberg constant from Bohr atom model:

a. Electron and proton is orbit: Coulomb force = centripedal force for stable orbit

where q_e is electron charge, m_e is electron mass, and e_o is a universal constant (the permittivity of the vacuum)

b. Bohr assumed angular momentum of electron orbits were in quantized (units of Planck's constant divided by 2p:

where n = integer (1,2,3,...)

c. The (Coulomb) energy of the electron in orbit n is

d. Combining a, b, c and solving for En gives

e. Using equation 3 above and solving for wavelength of the photon emitted by the transition of an electron from level n to level m:

or

as expected

5. The Stephan-Boltzmann Luminosity law. The luminosity (energy emitted per sec) per unit area for a 'black body' radiator is given by

where the constant s is:

For a spherical body (stars), the total luminosity is:

Note: This equation means that a relatively small increases in temperature produces a large change in luminosity.

Example 1: Suppose the surface temperature of the Sun increased by 25%. How would the luminosity change?

i.e., it would increase by x2.44 = 244%!

Example 2: Compare the total luminosities of the Sun (T=5800K, R = 700,000 km) and Sirius (T = 9800K, R = 1,200,000 km).

So Sirius is nearly 26x the luminosity of the Sun