Absolute magnitude

In astronomy, absolute magnitude ('M') is the apparent magnitude, 'm', that an object would have if it were at a distance of 10 parsecs (about 32,616 light-years, or 3 × 10¹⁴ km) in completely empty space with no interstellar absorption. The advantage of the absolute magnitude is that it has a direct relationship with the luminosities of the stars, being the same relationship for each one of them, thus being able, when comparing the absolute magnitudes between two or more stars, to also compare the luminosities between them — since the distance does not influence in any way.

Absolute magnitude of a comet or asteroid is the brightness that the star in question would have if it were located 1 AU from both the Sun and the Earth and its phase angle were 0°, that is, completely illuminated by the Sun.

The more luminous an object is, the smaller the numerical value of its absolute magnitude. A difference of 5 magnitudes between the absolute magnitudes of two objects corresponds to a ratio of 100 in their luminosities, and a difference of n magnitudes in absolute magnitude corresponds to a luminosity ratio of 100^n/5. For example, a star of absolute magnitude M_V = 3.0 would be 100 times more luminous than a star of absolute magnitude M_V = 8.0 measured in the V filter band. Sun has absolute magnitude M_V = +4.83. Very luminous objects can have negative absolute magnitudes: for example, the Milky Way galaxy has an absolute UBV photometric magnitude B of about −20.8.

The absolute bolometric magnitude of an object (M_bol) represents its total luminosity over all wavelengths, rather than in a single filter band, as It is expressed on a logarithmic magnitude scale. To convert an absolute magnitude in a specific filter band to an absolute bolometric magnitude, a bolometric correction (BC) is applied.

For Solar System bodies that shine in reflected light, a different definition of absolute magnitude (H) is used, based on a standard reference distance of one astronomical unit.

Definition

To define the absolute magnitude it is necessary to specify the type of electromagnetic radiation that is being measured. The absolute magnitude is generally derived from the visual magnitude measured with a filter V, and is expressed as M_v. If it is defined for other wavelengths, it will carry different subscripts, and if radiation at all wavelengths is considered, it is called the bolometric absolute magnitude (M_bol).

The absolute magnitude can be found, if the apparent magnitude is known (m{displaystyle m}) and distance (d{displaystyle d}) in parsec by means of:

M = m + 5 - 5 × log d [1]

If the parallax (π) is known, in seconds of arc, then we have:

M = m + 5 + 5 × log π [2]

For example, for Vega (α Lyr) it is m = +0.03 and π = 0.129”; having then:

M = 0.03 + 5 + (5 × 0.88941) = 0.58

one of a kind, is the Sun; its visual magnitude is m = –26.75, but the solar parallax is the one that corresponds to the astronomical unit of distance, which is contained 206264.806248 times in the parsec (1AU=1/206264.806248 pc), thus we will put this number of seconds, that is, π = 206264.806248”, with which

M = -26,75 + 5 + 5 × log 206264,806248 = -21,75 + 5 × 5,31443 = -21,75 + 26,57 = + 4.81

or:

M = -26,75 + 5 - 5 × log (1/206264,806248) = + 4.81

Bolometric magnitude

The bolometric magnitude, M_bol, takes into account electromagnetic radiation at all wavelengths. Includes those not observed due to instrumental passband, atmospheric absorption by Earth, and extinction by interstellar dust. It is defined based on the luminosity of the stars. For stars with few observations, it should be calculated assuming an effective temperature.

Classically, the difference in bolometric magnitude is related to the luminosity ratio according to:

Mborl,⋆ ⋆ − − Mborl,Δ Δ =− − 2,5log10 (L⋆ ⋆ LΔ Δ ){displaystyle M_{mathrm {bol,star } }-M_{mathrm {bol,odot } }=-2,5log _{10}left({frac {L_{star }}{L_{odot }}}}{right)}}

What it does per investment:

L⋆ ⋆ LΔ Δ =100,4(Mborl,Δ Δ − − Mborl,⋆ ⋆ ){displaystyle {frac {L_{star }}{L_{odot }=}=10^{0,4left(M_{mathrm {bol,odot } } }-M_{mathrm {bol,star }}{right}}}}

whereː

L_Δ is the lightness of the Sun (Bolometric lightness)

L_★ is the lightness of the star (bolometric lightness)

M_bol, is the bolometric magnitude of the Sun

M_Bol,★ is the bolometric magnitude of the star.

In August 2015, the International Astronomical Union passed Resolution B2 defining the zero points of the absolute and apparent bolometric magnitude scales in SI units for power (watts) and irradiance (W/m²), respectively. Although bolometric magnitudes had been used by astronomers for many decades, there were systematic differences in the absolute magnitude-luminosity scales presented in various astronomical references, and no international standardization. This led to systematic differences in the scales of bolometric corrections, which when combined with incorrectly assumed absolute bolometric magnitudes for the Sun could lead to systematic errors in estimated stellar luminosities (and calculated stellar properties that depend on stellar luminosity, such as radii)., ages and so on).

B2 resolution defines an absolute bolometric magnitude scale where M_bol = 0 corresponds to luminosity L₀ = 3.0128 × 10²⁸ W at zero point luminosity L₀ adjusted so that the Sun (with nominal luminosity 3.828 × 10²⁶ W) corresponds to the absolute bolometric magnitude M_bol,⊙ = 4.74. Placing a source of radiation (e.g. star) at the standard distance of 10 parsecs, it follows that the zero point of the apparent bolometric magnitude scale M _bol = 0 corresponds to the irradiation f0 = 2.518021002 × 10^-8 W/m². Using the UAI 2015 scale, the nominal total solar irradiance ("solar constant") measured in 1 astronomical unit (1361 W/m²) corresponds to an apparent bolometric magnitude of m_bol,⊙ = −26,832.

Following Resolution B2, the relationship between the absolute bolometric magnitude of a star and its luminosity is no longer directly tied to the (variable) luminosity of the Sun:

Mborl=− − 2,5log10 L⋆ ⋆ L0=− − 2,5log10 L⋆ ⋆ +71,197425...{displaystyle M_{mathrm {bol} }=-2,5log _{10}{frac {L_{star }{L_{0}}}}=-2,5log _{10}L_{star }+71,197425...}

whereː

L_★ is the luminosity of the star (bolometric lightness) in watts

L₀ is point zero luminosity 3.0128 × 10²⁸ W

M_bowl is the bolometric magnitude of the star

The new UAI absolute magnitude scale permanently disconnects the scale from the variable Sun. However, on this SI power scale, nominal solar luminosity closely corresponds to M _bol = 4.74, a value that was commonly adopted by astronomers prior to the 2015 UAI resolution.

The luminosity of the star in watts can be calculated based on its absolute bolometric magnitude M_bol as:

L⋆ ⋆ =L010− − 0,4MBorl{displaystyle L_{star }=L_{0}10^{-0.4M_{mathrm {Bol}}}}}}

using the variables defined above.