Stellar Magnitude

It seems counter-intuitive that a so-called magnitude 1 star is brighter than a magnitude 6 star. Hipparchus, who bequeathed Ptolemy and then us his classification system, looked upon his magnitude numbers as rankings according to star size (assuming that the latter accounted for brightness). Thus the brightest stars he referred to as being `of the first magnitude` and so on down to the just-about-visible stars, which he described as being `of the sixth magnitude`.

Additional magnitudes were later added, starting with Galileo who was able to make out otherwise invisible stars with his invention, the telescope. In 1856 the Oxford astronomer Norman Pogson devised a mathematical system (quickly adopted) that fitted Hipparchus's scheme. He proposed that a difference of five magnitudes be defined as a brightness ratio of exactly 100 to 1. Thus a difference of one magnitude between two sources corresponded to a brightness ratio between them of exactly the fifth root of 100 - a value (very close to 2.512) now known as Pogson's ratio. With this definition, it can easily be shown that the difference in the apparent magnitudes m and M of two stars can be expressed by the formula

m - M = -2.5 log (q/Q),

Where q and Q denote the 'quantities' of radiation measured from n and M respectively,

If the magnitude M star referred to above is a standard reference star, m for the other star can be thought of as an absolute magnitude value. In this case, the magnitude formula can be rewritten as

m = k - 2.5 log q,

where the calibrating constant k is determined by the measured quantity of light from the reference star (typically though not always Vega) and whichever magnitude value (typically though not always zero or near zero) one decides to allocate to that star. In practice, the value of k is modified by correctional factors to do with atmosphere, colour etc.

`QUANTITY` OF RADIATION

The rather vague term `quantity of radiation`, used above and represented by the letters q and Q in the formulae, alludes to one of several precisely defined quantities, typically energy flux (units W/m²), photon flux (photons/s/m²), or flux density (W/m²/Hz or Jy).

The terms `flux` (symbol F), `intensity` (I), and `apparent brightness` (b) are commonly used interchangeably in place of `energy flux` as in the preceding paragraph. Some authors have pointed out, though, that using the term `brightness` in this context is misleading. When dealing with an extended luminous object such as the Moon, the Sun, a nebula or a galaxy, the geometry of the situation means that the observed surface brightness is unlike the flux in that it is independent of the source-observer separation. Thus the red giant Betelgeuse only appears as bright as it does because it is a relatively extended object as stars go, subtending a much larger solid angle than most stars.

PASSBANDS

It is often desirable to allocate an individual magnitude to each passband for a given star. A passband is the wavelength range that can pass almost unattenuated through a specified filter. Photometry is the measurement and study of light. A `standard photometric system` is the conceptual and physical setup needed to make these measurements in a way that is repeatable and comparable across different locations. Such systems comprise: [1] a set of well-defined discrete passbands; and [2] a set of carefully measured primary standard stars. In the commonly used Johnson-Morgan, or UBV, system, for example, each of a set of main sequence stars, which includes Vega, has been assigned a magnitude of 0.03 for each of its three passbands (corresponding to UV, Blue and Visible). The system has been extended to incorporate two additional passbands (Red and IR). The corresponding passband magnitudes M are identified using subscri pts (U, B, V, R, I).

Note that a star which is `bright` in the visible spectrum (Sirius is the brightest) may be relatively-speaking 'dim' in another part of the spectrum, and vice versa. In the near-infrared J-band, it is Betelgeuse that turns out to be the 'brightest' star.

The magnitude formulae can now be written so as to take account of passband-specific measurements:

mp - Mp = -2.5 log (qp/Qp) and mp = kp - 2.5 log (qp)

Sometimes the bands themselves are given subscri pts to designate to a particular photometric system, e.g. Rj and Rc for the Red passband in the Johnson and Cousins systems respectively.

Differences between magnitudes in two passbands can often be measured more accurately for a star than individual absolute magnitudes can. Astrophysicists often refer to this difference for the star relative to the corresponding magnitude difference for, say, Vega.

SUBJECTIVE BRIGHTNESS

Suited as the logarithmic magnitude scale is to the quantification of objectively measurable quantities, it seems not so suitable for representing subjective 'brightness'. Recent experiments indicate that visual, auditory, etc. perceptions are characterised by power-law relationships in response to stimuli rather than the logarithmic ones. Thus a star that looks to the eye halfway in brightness between 2.0 and 4.0 will have a magnitude of about 2.8 rather than 3.0.