Definition of the differences in brightness was made over 2000 years ago when the Greek astronomer Hipparchus decided to classify stars by their brightness. He roughly divided stars into classes of brightness. Around A.D. 150 the Greek/Egyptian astronomer in Alexandria, Claudius Ptolemy, extended these classifications. In Ptolemy's classification, the brightest stars were said to be first magnitude stars. The next level of fainter stars were said to be second magnitude stars and so on down to the point when no fainter stars could be seen with the naked eye. The faintest stars seen were classified as sixth magnitude stars.

Stellar Magnitudes
During the 19th century astronomers began a more exact measurement of stellar brightnesses. Using the early system of stellar brightness, it was discovered that the brightest stars (first magnitude stars) were 100 times brighter than the faintest stars (sixth magnitude stars). That represents a five magnitude difference for 100 times brightness change. This range was then used to set up the current stellar magnitude system.

It turns out that some of the stars observed by the Greeks were actually much brighter than most of the first magnitude stars. This required the system to be extended to a number less than first magnitude. In fact it was necessary to extend it to zero magnitude and beyond into the negative magnitudes. This causes some confusion as the brighter the star the lesser its magnitude. In the case of stellar magnitudes, less is more. Brightness was also extended in the other direction. When telescopes were used, stars much fainter than the sixth magnitude were seen and needed to be classified.

Astronomers have used many different techniques to measure the brightness of stars. These include photgaphic means, measuring the density of star images on negatives, using a calibrated attenuating wedge to determine the point where the star could no longer be seen and techniques that involve the photoelectric effect.

Some simple math was used to determine the magnitude of stars between first and sixth magnitude. The division between first and sixth magnitude is 5 and the brightness change is 100. That relates to the fifth root of 100 for each change of magnitude ( 5100 = 2.512). Thus a second magnitude star is 2.512 times brighter than a third magnitude star and the third magnitude star 2.512 times brighter than a fourth magnitude star and so on.

If one star has a brightness B1 and another a brightness B2, the magnitude relationship is then

B1 / B2 = (10^2/5)^{(m2 - m1)}

or

log₁₀(B1 / B2) = 2/5(m2 - m1)

which can be rewritten as

m1 - m2 = -2.5 log₁₀ (B1 / B2)

Note:
An important point that may be confusing, the 2.5 is exactly 2.5 or (5/2) not 2.512 rounded off. The minus sign allows magnitudes for brighter stars to increase in a negative direction. Star with m= 3 is brighter than a star with m= 4 and a star with m= -2 is brighter than a star with m= 2.

To determine the magnitude of a single star (not the ratio between stars) the following equation is used:

M = - 2.5 log₁₀(B) + C

Where the M is the star's magnitude, B is a quantitative measurement of the star’s brightness (counts, current measurement, voltage measurement, deflection of a chart recorder) and C is a constant for the system used to adjust for the system sensitivity. The brightness of a given star in a six inch reflector telescope is much less than the brightness of the same star in a twelve inch reflector. The C takes that and other factors into account so measurements of the same star with different equipment will produce the same magnitude. There is still more to magnitudes.

Absolute Magnitude
The brightness of a star can be diminished by distance across space and atmospheric absorption (extinction). To standardize stellar magnitudes it was decided to use a system where the brightness of a star at a fixed distance from the star would be determined. This is known as the star's absolute magnitude. The absolute magnitude of a star is its brightness at a standard distance of 10 parsecs or 32.6 light years.

Apparent Magnitude
The apparent magnitude of a star is the brightness of the star if observed at the Earth, just outside the atmosphere. Most published magnitude values for stars are the apparent magnitude value. Thus when stars are measured on the ground beneath the atmosphere, the reduced brightness caused by the atmosphere must be considered.

Another factor for a star's brightness is the wavelength of the measured light. Most stars brighter than magnitude 9 have published magnitudes given in three bands, V, B and U (visual, blue and ultraviolet). Data for other bands is also published for certain stars. For example, Vega (alpha Lyra), has an absolute magnitude = 0.5 and apparent magnitudes V = 0.03, B = 0.03, U = 0.02 or Polaris (alpha Ursa Minor) with absolute magnitude = -17 and apparent magnitudes V = 2.02, B = 2.62 and U = 0.38.

Since most of us do not have the ability to observe outside the Earth's atmosphere, the starlight we see is dimmed by the absorption (known as atmospheric extinction) of light passing through the atmosphere. The extinction is least directly overhead (where the star light has the least atmosphere to penetrate) and greatest near the horizon. A given star may have a magnitude of V= 4.567 when measured directly overhead, but 4.986 when it approaches the horizon, yet the star's brightness is really constant. To make accurate brightness measurements of stars, this extinction must be figured in.

Counting Individual Photons
In 1887, Heinrich Rudolf Hertz (for whom the unit for frequency or number of cycles per second, Hertz or Hz, is named) observed that light from an electric spark falling on electrodes of another spark gap produced a lower breakdown voltage for that gap. In 1888 Wilhelm Hallwach discovered that ultraviolet light falling on a zinc plate caused a loss of negatives charge. Robert A. Millikan, among others, established the "laws" of photoelectric emission. One of the more important laws is "The number of photoelectrons emitted per second varies directly as the intensity of incident light." These developments laid the ground work and in 1905 Einstein formulated the photoelectric effect for which he was awarded the Nobel Prize in 1921. His theory explained the details of when a photon strikes the surface of some materials, an electron is ejected. This led to the development of photomultiplier tubes.

Photons have no charge and thus directly amplifying a single photon electronically is difficult or impossible. By using the photoelectric effect, it is possible to use the impact of a single photon to cause an electron to be ejected which through a series of electronic stages (nine or more stages) results in a pulse or cascade of electrons. This is exactly what a photomultiplier tube does. This pulse of electrons can be further amplified and measured.

In a photomultiplier tube the photocathode (where the light strikes) is negatively charged, typically at about -1000 VDC. There are several anodes at potentials progressively closer to ground potential or 0 VDC (i.e., progressively more positive). An incident photon hits the special photoemissive cathode (photocathode). The impact of the photon with the photocathode surface causes an electron to be ejected. Since the electron has a negative charge it is accelerated by the more positive voltage of the first anode. The accelerated electron hits the first anode causing multiple electrons to be emitted. These electrons, in turn, are accelerated and hit the second anode, which is even more positively charged, causing even more electrons to be emitted. This process may go on for several more times. A small pulse of current is then available at the output of the tube. Typically photomultiplier tubes have nine or more anodes with gains of over a million.

Interesting is the fact that during World War II one of the first photomultiplier tubes, the 931A, was used as a noise source. Also during the 1950's and 1960's the photomultiplier tubes found use for such a mundane purpose as an automatic headlight dimmer on plush automobiles. It may seem a long way from noise devices and headlight dimmers to a sensitive device used by astronomers to measure feeble starlight, but indeed that is the history. The same 931A tube (later the selected 931A or 1P21) is still used in some astronomical photoelectric photometers.