Apparent magnitude

La apparent magnitude (${displaystyle mathbb {m} }$) quantify the brightness of a star or celestial body observed from Earth. Consequently, the apparent magnitude depends on the luminosity of the object, the observer-object distance and the possible extinction of light caused by cosmic dust.

In the 2nd century B.C. C. the stars were cataloged by their apparent magnitude. For this, the visible stars were divided into six classes. The first class (m=+1) contained the brightest stars. In the sixth (m=+6), stars with very dim brightness were included. This classification is based on the response of the human eye to light; being this non-linear. Therefore, if there are three stars whose brightness follows the proportion 1:10:100, the third corresponds to the brightest, the difference between the apparent magnitudes of the first and the second is the same as the difference between the second and the third. This is because the scale is logarithmic.

Currently, the scale of apparent magnitudes can take any real value, both negative and positive, for example: the magnitude of the Sun is -26.7. Similarly, the apparent magnitude decreases when the brightness of the star increases, consequently, the apparent magnitude of the Sun is the lowest value. The Hubble Space Telescope makes it possible to observe objects with apparent magnitudes of +31.5.

The experimental measurement of the apparent magnitude of an object is based on photometry. This is why it is conditioned by the sensitivity of the instrument and its bandpass filter. Depending on the method of observation, different magnitude systems can be defined; among the most common are the UBV photometric system and the Strömgren photometric system.

Therefore, the apparent magnitude is measured for certain bands of the luminous spectrum. In the case of measuring in the visible spectrum, it is called visual magnitude (${displaystyle m_{v}}$) and can be estimated by the human eye. If measured in all wavelengths, it is called bolometric magnitude (${displaystyle m_{text{bol}}}$).

It is necessary to define a reference brightness value for the zero point of the apparent magnitude, usually Vega is used; other systems are STMAG and AB.

History

Norman Pogson known for formalizing the scale of apparent magnitudes.

The scale with which the apparent magnitude is measured originates from the Hellenistic practice of dividing the stars visible with the naked eye (without the aid of a telescope) into the interval from 1 to 6. The most visible stars were part of the first magnitude. While the weakest were considered in the sixth magnitude, this being the limit of human visual perception. This method of indicating the visibility of the stars with the naked eye was disclosed by Claudius Ptolemy in his Almagest, and it is believed that it may have been originated by Hipparchus of Nicaea. This system did not measure the magnitude of the Sun and considered that between one magnitude (m) and the next (m+1) the brightness doubled.

In 1856, Norman Pogson formalized the scale system defining that a star of first magnitude is 100 times brighter than a star of sixth magnitude. Thus, a star of first magnitude is ${displaystyle {sqrt[{5}}{100}}}approx 2.512}$ sometimes more visible than a second magnitude; remembering that it is a logarithmic scale. This value is known as the quotient of Pogson. The Pogson scale was originally set assigning to the brightness of the star Polaris the magnitude 2. However, it has been discovered that the Polar star is slightly variable, so currently the shine of the Vega star is used as a reference for the zero point of the apparent magnitude in any wavelength.

Mathematical formulation

Apparent magnitude ${displaystyle m_{x}}$ in the band ${displaystyle x}$ is defined as:

${displaystyle m_{x}=-2,5log _{10}left({frac {F_{x}}{F_{x,0}}}}}{F_{x,0}}}}}}}$ [1]

where ${displaystyle F_{x}}$ is the luminous flux observed in the band ${displaystyle x}$ and ${displaystyle F_{x,0}}$ is the reference flow for point zero of the apparent magnitude.

It can be verified that the definition of the apparent magnitude contains the formalization imposed by Pogson, in which for stars with magnitudes ${displaystyle m}$, ${displaystyle m+1}$ and flows ${displaystyle F_{m}}$, ${displaystyle F_{m+1}}$ respectively:

${displaystyle m-(m+1)=-2,5log _{10}left({frac {F_{m}{F_{0}}}}right)+2,5log _{10}{left({frac}{F_{m+1}}{F_{F_{0}}}{FFFFF}{cHFFFFFF}{$

${displaystyle Rightarrow {frac {F_{m}}{F_{m+1}}}}=10^{frac {1}{2,5}}}={sqrt[{5}{100}}}}{approx 2,512}$

Likewise it is fulfilled that a star of magnitude ${displaystyle m}$ is 100 times brighter than a star of magnitude ${displaystyle m+5}$.

Systems of Magnitude

Different magnitude systems are used to determine a reference point (flux) for the zero value of the apparent magnitude. The most common magnitude systems are introduced below.

• VegaMAG: It is taken as reference brightness to the Vega star in any wavelength, therefore ${displaystyle m_{text{Vega}}=0}$. Vega is observable for more than 6 months (on the northern horizon) and has a relatively mild spectral distribution of energy. However, systems based on a particular star have now been no longer considered because of the stability of the star. This is why in current systems it is found that Vega's apparent magnitude is 0.03 in the visual spectrum.
• ABMAG and STMAG: These are flow-based systems. STMAG is set in a spectrum with constant flow density per wavelength unit (${displaystyle F_{lambda }$), while ABMAG is set in a spectrum with constant flow density per frequency unit (${displaystyle F_{nu }}$). They are expressed in the form:

${displaystyle m_{ST}=-2.5log _{10}left(F_{lambda }right)-21.1}$

${displaystyle m_{AB}=-2.5log _{10}left(F_{nu }right)-48.6}$

Where ${displaystyle F_{lambda }$ is in units ${displaystyle {text{erg}};s^{-1}cm^{-2}{text{{{Å}}{{}}{{-1}}}{displaystyle {text{text}}}}}}{{{text{{{}}}{{{}}}}{$ and ${displaystyle F_{nu }}$ is in units ${displaystyle {text{erg}};s^{-1}cm^{-2}{text{Hz}}{-1}}}{{text}}{1}}{cHz}}}}$. For both cases the zero point is considered such that the magnitude of Vega in these systems coincides with the visible spectrum.

Pogson's Law

In general, the difference between two apparent magnitudes ${displaystyle m_{1}}$ and ${displaystyle m_{2}}$, with their respective flows in the same band is obtained from:

${displaystyle m_{1}-m_{2}=-2,5log _{10}left({frac {F_{m_{1}}}}{F_{m_{2}}}}}}}}{F_{F_{m_{2}}}}}}}$

This relationship allows for a relative comparison between the brightness of two objects and is known as Pogson's Law.

Other magnitudes

Absolute magnitude

From the apparent magnitude and distance ${displaystyle d}$ (measured) of a star it is possible to determine its absolute magnitude ${displaystyle M}$. This being the magnitude that would have if the star was at a distance of 10 pc:

${displaystyle m-M=5log _{10}left({frac {d}{10}}}right}$

This relationship is obtained considering that the flow of an object is inversely proportional to the square of its distance. In addition, light rays from the star usually go through a phenomenon of extinction; caused by particles (poor) located on the path of lightning. Therefore, it is necessary to take into account the loss by cosmic dust by introducing correction ${displaystyle A}$ in magnitude, such as:

${displaystyle m-M=5log _{10}left({frac {d}{10}}}}right)+A}$

Likewise, it is possible to introduce a correction due to redshift and movement between the object and the reference system; called K correction.

Bolometric magnitude

The bolometric magnitude ${displaystyle m_{text{bol}}}$ (appears) takes into account the radiated flow in any wavelength and is defined as:

${displaystyle m_{text{bol}}=-2,5log _{10}left(int _{0}^{infty }F_{nu },dnu right)+C_{text{bol}}}}$

Where ${displaystyle C_{text{bol}}}$ defines the zero point of the bolometric magnitude.

It is possible to know the absolute bolometric magnitude ${displaystyle M_{text{bol}}}$ from the luminosity ${displaystyle L}$ of the star and taking the Sun as such reference point you get:

${displaystyle M_{text{bol}}-M_{{text{{bol}}}odot }=-2.5log _{10}left({frac {L}{L_{odot }}}}{right}$

Being ${displaystyle M_{text{bol}}odot }}$ = +4.74 and ${displaystyle L_{odot }$ = 3.0128×10²⁸ W the absolute bolometric magnitude and the luminosity of the Sun respectively.