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# Apparent magnitude

The apparent magnitude (m) of a star, planet or other heavenly body is a measure of its apparent brightness; that is, its brightness without regard to the object's distance from a point of observation. The dimmer an object appears, the higher its apparent magnitude. Note that apparent brightness is not equal to actual brightness - an extremely bright object may appear quite dim, if it is far away. The rate at which apparent brightness changes, as the distance from an object increases, is calculated by the inverse-square law. The absolute magnitude, M, of an object, is the apparent magnitude it would have if it were 10 parsecs away.

Scale of apparent magnitudes
App. Mag.Celestial Object
-26.8Sun
-12.6full Moon
-4.4Maximum brightness of Venus
-2.8Maximum brightness of Mars
-1.5Brightest star: Sirius
-0.7Second brightest star: Canopus
+6.0Faintest stars observable with naked eye
+12.6Brightest quasar
+30Faintest objects observable
with Hubble Space Telescope

The scale, upon which magnitude is measured, has its roots in the Hellenistic practice of dividing those stars visible to the naked eye into six magnitudes. The brightest stars were said to be of first magnitude (m = +1), those which were only half as bright were of second magnitude, and so on up to sixth magnitude (m = +6), the limit of human visual perception (without a telescope or the like). This somewhat crude method of indicating the brightness of stars was popularized by Ptolemy in his Almagest, and is generally believed to have originated with Hipparchus. This original system did not measure the magnitude of Sol.

In 1856, Pogson formalized the system by defining a typical first magnitude star as a star which is 100 times brighter than a typical sixth magnitude star; thus, a first magnitude star is 2.512 times brighter than a second magnitude star. The fifth root of 100 (2.512) is known as Pogson's Ratio. Pogson's scale was originally fixed by assigning Polaris a magnitude of 2. Astronomers have since discovered that Polaris is slightly variable -- Vega is the standard reference star.

The modern system is no longer limited to 6 magnitudes. Really bright objects have negative magnitudes. For example, Sirius, the brightest star of the celestial sphere, has an apparent magnitude of -1.44 to -1.46. The modern scale includes Luna and Sol, Luna has an apparent magnitude of -12.6 and Sol has an apparent magnitude of -26.7. The Hubble and Keck telescopes have located stars with magnitudes of +30.

The apparent magnitude in the band x can be defined as

where Fx is the observed flux in the band x, and C is a constant that depends in the units of the flux and the band.

The second thing to notice is that the scale is logarithmic: the relative brightness of two objects is determined by the difference of their magnitudes. For example, a difference of 3.2 means that one object is about 19 times as bright as the other, because Pogson's ratio raised to the power 3.2 is 19.054607... The logarithmic nature of the scale is due to the fact of the human eye itself having a logarithmic response, see Weber-Fechner Law.

Magnitude is complicated by the fact that light is not monochromatic. The sensitivity of a light detector varies according to the wavelength of the light, and the way in which it varies depends on the type of light detector. For this reason, it is necessary to specify how the magnitude is measured in order for the value to be meaningful. For this purpose the UBV system is widely used, in which the magnitude is measured in three different wavelength bands: U (centred at about 350 nm, in the near ultraviolet), B (about 435 nm, in the blue region) and V (about 555 nm, in the middle of the human visual range). The V band was chosen so that it gives magnitudes closely corresponding to those seen by the human eye, and when an apparent magnitude is given without any further qualification, it is usually the V magnitude that is meant, also called visual magnitude.

Since cooler stars, such as red giants and red dwarfs, emit little energy in the blue and UV reaches of the spectrum their power is often under-represented by the UBV scale. Indeed, some L and T class stars would have a UBV magnitude of well over 100 since they emit extremely little visible light, but are strongest in infra-red.

Magnitude is a minefield and it is extremely important to measure like with like. On photographic film, the relative brightnesses of the blue supergiant Rigel and the red supergiant Betelgeuse are reversed compared to what our eyes see since film is more sensitive to blue light than it is to red light.

For an object with given absolute magnitude, 5 is added to the relative magnitude when the distance is multiplied by 10.

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