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1. Doppler Shift with Sound Waves - Circular Orbits:

The received frequency changes if the source of sound moves towards or away from an observer. This is the Doppler Effect, demonstrated by Professor Lewin with a tuning fork. The fractional change in frequency reveals the velocity component along your line of sight to the moving sound source. If the source of sound is in circular motion, and if the observer is somewhere in the orbital plane, you can determine the orbital radius and the speed of the source in its orbit. This is demonstrated with a rotating wind organ.

2. Doppler Shift of Electromagnetic Radiation:

Electromagnetic radiation travels at the speed of light, c, in vacuum. If a source of light has a velocity component towards you, the frequencies that you will observe will be higher than those of the emitted radiation, and the received wavelengths will be shorter (blue-shift) than the emitted wavelengths. If the source is receding from you the received wavelength is longer (red-shift). The spectroscopic Doppler shift is used by astronomers to measure the radial velocity of emitters and absorbers of light.

3. Star Mass Determinations from Doppler Shift:

A binary star system consists of a pair of stars orbiting about their center of mass. By measuring the Doppler shifts of both stars as a function of time, you can determine the orbital period, the radial velocity of each star and, if the observer is located in the orbital plane, the orbital radii can be found for both stars. The orbital radii and Kepler's third Law determine the total mass of the system, enabling the determination of each star's mass separately.

4. X-ray Binary Systems:

In an X-ray binary system, there is a neutron star (or black hole) pulling matter off its donor companion. Matter spirals toward the neutron star, and potential energy is converted to kinetic energy. This, coupled with the high mass transfer rate between the pair, generates tremendous power and astronomical temperatures (it radiates mainly X-rays). The accreting ionized matter gets funnelled onto hot spots by the neutron star's magnetic field, which spins with the neutron star (making it an X-ray pulsar). Doppler shifts in the pulsar period and X-ray eclipses can provide orbital parameters (and masses) for the stellar system.

5. Black Holes:

A black hole is a massive object with no size, but with a characteristic surface called the event horizon from within which nothing can escape the black hole. In black-hole binary systems the accreting matter radiates X-rays as it approaches the event horizon of the black hole, but the black hole has no surface, therefore does not exhibit a pulsar-like behavior. You can measure the optical Doppler shift of the donor star, and from its spectrum estimate the donor's mass. This then leads to the mass of the accretor. If this mass is in excess of about 3 solar masses, it is believed to be a black hole. Cygnus X-1 was the first such discovery (in 1972). Its black hole is about 10 solar masses.