Stars and galaxies (M)
In stellar and galactic astronomy, the standard distance is 10 parsecs (about 32.616 light-years, 308.57 petameters or 308.57 trillion kilometres). A star at 10 parsecs has a parallax of 0.1″ (100 milliarcseconds). Galaxies (and other extended objects) are much larger than 10 parsecs, their light is radiated over an extended patch of sky, and their overall brightness cannot be directly observed from relatively short distances, but the same convention is used. A galaxy's magnitude is defined by measuring all the light radiated over the entire object, treating that integrated brightness as the brightness of a single point-like or star-like source, and computing the magnitude of that point-like source as it would appear if observed at the standard 10 parsecs distance. Consequently, the absolute magnitude of any object equals the apparent magnitude it would have if it were 10 parsecs away.
The measurement of absolute magnitude is made with an instrument called a bolometer. When using an absolute magnitude, one must specify the type of electromagnetic radiation being measured. When referring to total energy output, the proper term is bolometric magnitude. The bolometric magnitude usually is computed from the visual magnitude plus a bolometric correction, Mbol = MV + BC. This correction is needed because very hot stars radiate mostly ultraviolet radiation, whereas very cool stars radiate mostly infrared radiation (see Planck's law).
Some stars visible to the naked eye have such a low absolute magnitude that they would appear bright enough to outshine the planets and cast shadows if they were at 10 parsecs from the Earth. Examples include Rigel (−7.0), Deneb (−7.2), Naos (−6.0), and Betelgeuse (−5.6). For comparison, Sirius has an absolute magnitude of 1.4, which is brighter than the Sun, whose absolute visual magnitude is 4.83 (it actually serves as a reference point). The Sun's absolute bolometric magnitude is set arbitrarily, usually at 4.75.
Absolute magnitudes of stars generally range from −10 to +17. The absolute magnitudes of galaxies can be much lower (brighter). For example, the giant elliptical galaxy M87 has an absolute magnitude of −22 (i.e. as bright as about 60,000 stars of magnitude −10).
The Greek astronomer Hipparchus established a numerical scale to describe the brightness of each star appearing in the sky. The brightest stars in the sky were assigned an apparent magnitude m = 1, and the dimmest stars visible to the naked eye are assigned m = 6. The difference between them corresponds to a factor of 100 in brightness. For objects within the immediate neighborhood of the Sun, the absolute magnitude M and apparent magnitude m from any distance d (in parsecs) is related by:
where F is the radiant flux measured at distance d (in parsecs), F10 the radiant flux measured at distance 10 pc. The relation can be written in terms of logarithm:
where the insignificance of extinction by gas and dust is assumed. Typical extinction rates within the galaxy are 1 to 2 magnitudes per kiloparsec, when dark clouds are taken into account.
For objects at very large distances (outside the Milky Way) the luminosity distance dL must be used instead of d (in parsecs), because the Euclidean approximation is invalid for distant objects and general relativity must be taken into account. Moreover, the cosmological redshift complicates the relation between absolute and apparent magnitude, because the radiation observed was shifted into the red range of the spectrum. To compare the magnitudes of very distant objects with those of local objects, a K correction might have to be applied to the magnitudes of the distant objects.
The absolute magnitude M can also be approximated using apparent magnitude m and stellar parallax p:
or using apparent magnitude m and distance modulus μ:
Rigel has a visual magnitude mV of 0.12 and distance about 860 light-years
Vega has a parallax p of 0.129″, and an apparent magnitude mV of 0.03
Alpha Centauri A has a parallax p of 0.742″ and an apparent magnitude mV of −0.01
The Black Eye Galaxy has a visual magnitude mV of 9.36 and a distance modulus μ of 31.06
The bolometric magnitude Mbol, takes into account electromagnetic radiation at all wavelengths. It includes those unobserved due to instrumental pass-band, the Earth's atmospheric absorption, and extinction by interstellar dust. It is defined based on the luminosity of the stars. In the case of stars with few observations, it must be computed assuming an effective temperature.
Classically, the difference in bolometric magnitude is related to the luminosity ratio according to:
which makes by inversion:
- L⊙ is the Sun's luminosity (bolometric luminosity)
- L★ is the star's luminosity (bolometric luminosity)
- Mbol,⊙ is the bolometric magnitude of the Sun
- Mbol,★ 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 had been systematic differences in the absolute magnitude-luminosity scales presented in various astronomical references, and no international standardization. This led to systematic differences in bolometric corrections scales. Combined with incorrect assumed absolute bolometric magnitudes for the Sun could lead to systematic errors in estimated stellar luminosities (and stellar properties calculated which rely on stellar luminosity, such as radii, ages, and so on).
Resolution B2 defines an absolute bolometric magnitude scale where Mbol = 0 corresponds to luminosity L0 = ×1028 W, with the zero point 3.0128luminosity L0 set such that the Sun (with nominal luminosity ×1026 W) corresponds to absolute 3.828bolometric magnitude Mbol,⊙ = 4.74. Placing a radiation source (e.g. star) at the standard distance of 10 parsecs, it follows that the zero point of the apparent bolometric magnitude scale mbol = 0 corresponds to irradiance f0 = 021002×10−8 W/m2. Using the IAU 2015 scale, the nominal total 2.518solar irradiance ("solar constant") measured at 1 astronomical unit (2) corresponds to an apparent 1361 W/mbolometric magnitude of the Sun of mbol,⊙ = −26.832.
Following Resolution B2, the relation between a star's absolute bolometric magnitude and its luminosity is no longer directly tied to the Sun's (variable) luminosity:
- L★ is the star's luminosity (bolometric luminosity) in watts
- L0 is the zero point luminosity ×1028 W3.0128
- Mbol is the bolometric magnitude of the star
The new IAU absolute magnitude scale permanently disconnects the scale from the variable Sun. However, on this SI power scale, the nominal solar luminosity corresponds closely to Mbol = 4.74, a value that was commonly adopted by astronomers before the 2015 IAU resolution.
The luminosity of the star in watts can be calculated as a function of its absolute bolometric magnitude Mbol as:
using the variables as defined previously.