TheSun has an intrinsic luminosity of3.83×1026watts. In astronomy, this amount is equal to onesolar luminosity, represented by the symbolL⊙. A star with four times the radiative power of the Sun has a luminosity of4 L⊙.
InSI units, luminosity is measured injoules per second, orwatts. In astronomy, values for luminosity are often given in the terms of theluminosity of the Sun,L⊙. Luminosity can also be given in terms of the astronomicalmagnitude system: theabsolute bolometric magnitude (Mbol) of an object is a logarithmic measure of its total energy emission rate, whileabsolute magnitude is a logarithmic measure of the luminosity within some specificwavelength range orfilter band.
In contrast, the termbrightness in astronomy is generally used to refer to an object's apparent brightness: that is, how bright an object appears to an observer. Apparent brightness depends on both the luminosity of the object and the distance between the object and observer, and also on anyabsorption of light along the path from object to observer.Apparent magnitude is a logarithmic measure of apparent brightness. The distance determined by luminosity measures can be somewhat ambiguous, and is thus sometimes called theluminosity distance.
When not qualified, the term "luminosity" means bolometric luminosity, which is measured either in theSI units,watts, or in terms ofsolar luminosities (L☉). Abolometer is the instrument used to measureradiant energy over a wide band byabsorption and measurement of heating. A star also radiatesneutrinos, which carry off some energy (about 2% in the case of the Sun), contributing to the star's total luminosity.[5] The IAU has defined a nominal solar luminosity of3.828×1026 W to promote publication of consistent and comparable values in units of the solar luminosity.[6]
While bolometers do exist, they cannot be used to measure even the apparent brightness of a star because they are insufficiently sensitive across theelectromagnetic spectrum and because most wavelengths do not reach the surface of the Earth. In practice bolometric magnitudes are measured by taking measurements at certain wavelengths and constructing a model of the total spectrum that is most likely to match those measurements. In some cases, the process of estimation is extreme, with luminosities being calculated when less than 1% of the energy output is observed, for example with a hotWolf-Rayet star observed only in the infrared. Bolometric luminosities can also be calculated using abolometric correction to a luminosity in a particular passband.[7][8]
The term luminosity is also used in relation to particularpassbands such as a visual luminosity ofK-band luminosity.[9] These are not generally luminosities in the strict sense of an absolute measure of radiated power, but absolute magnitudes defined for a given filter in aphotometric system. Several different photometric systems exist. Some such as the UBV orJohnson system are defined against photometric standard stars, while others such as theAB system are defined in terms of aspectral flux density.[10]
A star's luminosity can be determined from two stellar characteristics: size andeffective temperature.[11] The former is typically represented in terms of solarradii,R⊙, while the latter is represented inkelvins, but in most cases neither can be measured directly. To determine a star's radius, two other metrics are needed: the star'sangular diameter and its distance from Earth. Both can be measured with great accuracy in certain cases, with cool supergiants often having large angular diameters, and some cool evolved stars havingmasers in their atmospheres that can be used to measure the parallax usingVLBI. However, for most stars the angular diameter or parallax, or both, are far below our ability to measure with any certainty. Since the effective temperature is merely a number that represents the temperature of a black body that would reproduce the luminosity, it obviously cannot be measured directly, but it can be estimated from the spectrum.
An alternative way to measure stellar luminosity is to measure the star's apparent brightness and distance. A third component needed to derive the luminosity is the degree ofinterstellar extinction that is present, a condition that usually arises because of gas and dust present in theinterstellar medium (ISM), theEarth's atmosphere, andcircumstellar matter. Consequently, one of astronomy's central challenges in determining a star's luminosity is to derive accurate measurements for each of these components, without which an accurate luminosity figure remains elusive.[12] Extinction can only be measured directly if the actual and observed luminosities are both known, but it can be estimated from the observed colour of a star, using models of the expected level of reddening from the interstellar medium.
In the current system ofstellar classification, stars are grouped according to temperature, with the massive, very young and energeticClass O stars boasting temperatures in excess of 30,000 K while the less massive, typically olderClass M stars exhibit temperatures less than 3,500 K. Because luminosity is proportional to temperature to the fourth power, the large variation in stellar temperatures produces an even vaster variation in stellar luminosity.[13] Because the luminosity depends on a high power of the stellar mass, high mass luminous stars have much shorter lifetimes. The most luminous stars are always young stars, no more than a few million years for the most extreme. In theHertzsprung–Russell diagram, the x-axis represents temperature or spectral type while the y-axis represents luminosity or magnitude. The vast majority of stars are found along themain sequence with blue Class O stars found at the top left of the chart while red Class M stars fall to the bottom right. Certain stars likeDeneb andBetelgeuse are found above and to the right of the main sequence, more luminous or cooler than their equivalents on the main sequence. Increased luminosity at the same temperature, or alternatively cooler temperature at the same luminosity, indicates that these stars are larger than those on the main sequence and they are called giants or supergiants.
Blue and white supergiants are high luminosity stars somewhat cooler than the most luminous main sequence stars. A star likeDeneb, for example, has a luminosity around 200,000L⊙, a spectral type of A2, and an effective temperature around 8,500 K, meaning it has a radius around 203 R☉ (1.41×1011m). For comparison, the red supergiantBetelgeuse has a luminosity around 100,000L⊙, a spectral type of M2, and a temperature around 3,500 K, meaning its radius is about 1,000 R☉ (7.0×1011m). Red supergiants are the largest type of star, but the most luminous are much smaller and hotter, with temperatures up to 50,000 K and more and luminosities of several millionL⊙, meaning their radii are just a few tens ofR⊙. For example,R136a1 has a temperature over 46,000 K and a luminosity of more than 6,100,000L⊙[14] (mostly in the UV), it is only 39 R☉ (2.7×1010m).
The luminosity of aradio source is measured inW Hz−1, to avoid having to specify abandwidth over which it is measured. The observed strength, orflux density, of a radio source is measured inJansky where1 Jy = 10−26 W m−2 Hz−1.
For example, consider a 10W transmitter at a distance of 1 million metres, radiating over a bandwidth of 1 MHz. By the time that power has reached the observer, the power is spread over the surface of a sphere with area4πr2 or about1.26×1013 m2, so its flux density is10 / 106 / (1.26×1013) W m−2 Hz−1 = 8×107 Jy.
More generally, for sources at cosmological distances, ak-correction must be made for the spectral index α of the source, and a relativistic correction must be made for the fact that the frequency scale in the emittedrest frame is different from that in the observer'srest frame. So the full expression for radio luminosity, assumingisotropic emission, iswhereLν is the luminosity inW Hz−1,Sobs is the observedflux density inW m−2 Hz−1,DL is theluminosity distance in metres,z is the redshift,α is thespectral index (in the sense, and in radio astronomy, assuming thermal emission the spectral index is typicallyequal to 2.)[15]
For example, consider a 1 Jy signal from a radio source at aredshift of 1, at a frequency of 1.4 GHz.Ned Wright's cosmology calculator calculates aluminosity distance for a redshift of 1 to be 6701 Mpc = 2×1026 m giving a radio luminosity of10−26 × 4π(2×1026)2 / (1 + 1)(1 + 2) = 6×1026 W Hz−1.
To calculate the total radio power, this luminosity must be integrated over the bandwidth of the emission. A common assumption is to set the bandwidth to the observing frequency, which effectively assumes the power radiated has uniform intensity from zero frequency up to the observing frequency. In the case above, the total power is4×1027 × 1.4×109 = 5.7×1036 W. This is sometimes expressed in terms of the total (i.e. integrated over all wavelengths) luminosity of theSun which is3.86×1026 W, giving a radio power of1.5×1010L⊙.
Point sourceS is radiating light equally in all directions. The amount passing through an areaA varies with the distance of the surface from the light.
TheStefan–Boltzmann equation applied to ablack body gives the value for luminosity for a black body, an idealized object which is perfectly opaque and non-reflecting:[11]whereA is the surface area,T is the temperature (in kelvins) andσ is theStefan–Boltzmann constant, with a value of5.670374419...×10−8 W⋅m−2⋅K−4.[16]
Imagine a point source of light of luminosity that radiates equally in all directions. A hollowsphere centered on the point would have its entire interior surface illuminated. As the radius increases, the surface area will also increase, and the constant luminosity has more surface area to illuminate, leading to a decrease in observed brightness.
The surface area of a sphere with radiusr is, so for stars and other point sources of light:where is the distance from the observer to the light source.
Luminosity is an intrinsic measurable property of a star independent of distance. The concept of magnitude, on the other hand, incorporates distance. Theapparent magnitude is a measure of the diminishing flux of light as a result of distance according to theinverse-square law.[17] The Pogson logarithmic scale is used to measure both apparent and absolute magnitudes, the latter corresponding to the brightness of a star or othercelestial body as seen if it would be located at an interstellar distance of 10parsecs (3.1×1017metres). In addition to this brightness decrease from increased distance, there is an extra decrease of brightness due to extinction from intervening interstellar dust.[18]
By measuring the width of certain absorption lines in thestellar spectrum, it is often possible to assign a certain luminosity class to a star without knowing its distance. Thus a fair measure of its absolute magnitude can be determined without knowing its distance nor the interstellar extinction.
In measuring star brightnesses, absolute magnitude, apparent magnitude, and distance are interrelated parameters—if two are known, the third can be determined. Since the Sun's luminosity is the standard, comparing these parameters with the Sun's apparent magnitude and distance is the easiest way to remember how to convert between them, although officially, zero point values are defined by the IAU.
The magnitude of a star, aunitless measure, is a logarithmic scale of observed visible brightness. The apparent magnitude is the observed visible brightness fromEarth which depends on the distance of the object. The absolute magnitude is the apparent magnitude at a distance of 10 pc (3.1×1017m), therefore the bolometric absolute magnitude is a logarithmic measure of the bolometric luminosity.
The difference in bolometric magnitude between two objects is related to their luminosity ratio according to:[19]
where:
is the bolometric magnitude of the first object
is the bolometric magnitude of the second object.
is the first object's bolometric luminosity
is the second object's bolometric luminosity
The zero point of the absolute magnitude scale is actually defined as a fixed luminosity of3.0128×1028 W. Therefore, the absolute magnitude can be calculated from a luminosity in watts:whereL0 is the zero point luminosity3.0128×1028 W
and the luminosity in watts can be calculated from an absolute magnitude (although absolute magnitudes are often not measured relative to an absolute flux):
^Mamajek, E. E.; Prsa, A.; Torres, G.; Harmanec, P.; Asplund, M.; Bennett, P. D.; Capitaine, N.; Christensen-Dalsgaard, J.; Depagne, E.; Folkner, W. M.; Haberreiter, M.; Hekker, S.; Hilton, J. L.; Kostov, V.; Kurtz, D. W.; Laskar, J.; Mason, B. D.; Milone, E. F.; Montgomery, M. M.; Richards, M. T.; Schou, J.; Stewart, S. G. (2015). "IAU 2015 Resolution B3 on Recommended Nominal Conversion Constants for Selected Solar and Planetary Properties".arXiv:1510.07674 [astro-ph.SR].
^Delfosse, Xavier; et al. (December 2000), "Accurate masses of very low mass stars. IV. Improved mass-luminosity relations",Astronomy and Astrophysics,364:217–224,arXiv:astro-ph/0010586,Bibcode:2000A&A...364..217D
^Doran, E. I.; Crowther, P. A.; de Koter, A.; Evans, C. J.; McEvoy, C.; Walborn, N. R.; Bastian, N.; Bestenlehner, J. M.; Gräfener, G.; Herrero, A.; Kohler, K.; Maiz Apellaniz, J.; Najarro, F.; Puls, J.; Sana, H.; Schneider, F. R. N.; Taylor, W. D.; van Loon, J. Th.; Vink, J. S. (2013). "The VLT-FLAMES Tarantula Survey - XI. A census of the hot luminous stars and their feedback in 30 Doradus".Astronomy & Astrophysics.558: A134.arXiv:1308.3412v1.Bibcode:2013A&A...558A.134D.doi:10.1051/0004-6361/201321824.S2CID118510909.
