Definition magnitudes

The first classifications of star brightness are due to the Greeks in particular with Ptolémée. Classification rested on the observations of stars in the twilight; first stars appearing being most brilliant, the last least brilliant. The twilight was divided into 6 equal periods of time making it possible to gather stars in 6 groups magnitudes. It is in particular for this reason that the stars of weak luminosity have the largest magnitudes.

The magnitudes thus correspond to the perception of the luminosity by the eye, it is not linear: we do not perceive the luminosity of stars but rather their differences relative. The scale magnitudes observed is thus logarithmic curve.

It is into 1856 that Norman Pogson (pogson1856) defined in a formal way the magnitudes to make it possible to reproduce the astronomical observations in the form:

$$m = -2.5\log_{10}(E) + C$$ (A.1)

This law is known under the name of formula of Pogson, the factor $2.5$ is the scale of Pogson, E the brightness of star and C an arbitrary constant: zero point.

This law defines in fact a scale, it is standardized using a star which defines the system magnitude.

In this system, the magnitude of star of reference has by definition a magnitude 0. One thus can récrire the formula of Pogson like:

$$m = -2.5\log_{10}(E_{*}/E_{ref})$$ (A.2)

where $E_{ref}$ are respectively brightnesses of the object considered and star of reference. In flux, this expression gives:

$$m = -2.5\log_{10}(f_{*}/f_{ref})$$ (A.3)

where $f_{ref}$ are respectively fluxes of the object considered and star of reference.

Item zero thus can récrire like:

$$zero = 2.5\log_{10}(f_{ref})$$ (A.4)

and thus magnitude:

$$m = -2.5\log_{10}(f_{*}) + zero$$ (A.5)

The flux actually observed of a source depends on several parameters:

1. The band-width of the filters which are used at the time of the observations: $T_f(\lambda)$.

2. Transmission of the optical system (reflectivity of the mirrors, transmission of the various optical correctors...): $T_{opt}(\lambda)$.

3. Quantum effectiveness of CCDs: $Qe(\lambda)$.

4. Transmission of the atmosphere $T_{atm}$ and quantity of crossed atmosphere: airmass X.

The flux of a star can thus be written like the radiance of star $I(\lambda)$ multiplied by the system of observation:

$$f = \int_{0}^{+\infty}T_{atm}(\lambda,X)~T_{opt}(\lambda)~Qe(\lambda)~T_f(\lambda) ~I(\lambda) d\lambda$$ (A.6)

And by definition zero point is:

$$zero = 2.5\log_{10}\left(\int_{-\infty}^{+\infty}T_{atm}(\lambda,X)~T_{opt}(\lambda)~Qe(\lambda)~T_f(\lambda) ~I_{ref}(\lambda) d\lambda \right)$$ (A.7)

where $I_{ref}(\lambda)$ is radiance according to the wavelength of star of reference.

The transmission of optics, the quantum effectiveness of the CCD and the transmission are characteristics of the instrument and are constant in the course of time, they make it possible to define zero point except atmosphere.

On the other hand, atmospheric absorption and the airmass crossed are a function of the climatic conditions and the angle of sight compared to the zenith. They must thus be given for each observation.

The star of reference of the standard system magnitude more the current is Vega ($\alpha-$lyrae ), the observation of this star by the largest telescopes is impossible bus too luminous.

The measurement of absorption is thus made thanks to the star observation of standard catalogues, in a regular way (several times per night).

The terms of correction of airmass are as for them given by using the standard star observations with various values of airmass.

Lastly, to be comparable with others, a magnitude must be expressed in the system of standard filter. In practice, the transmission of the instruments of observation is only one approximation of these filters, of the differences of the order of the percent are measurable. In particular, the stars having colors very different from star of reference will see their magnitude measured in these instruments biased.