2001:5A8:4613:AF00:6D58:CE16:7D8C:50CC: /* Schechter luminosity function */Fixed mistaken edit.

2024-03-01T18:57:38Z

Schechter luminosity function: Fixed mistaken edit.

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{{For|the luminosity function in photometry|Luminosity function}}
In [[astronomy]], a '''luminosity function''' gives the number of [[star]]s or [[galaxy|galaxies]] per [[luminosity]] interval.<ref name="stahler">{{cite book | author1 = Stahler, S. |author2=Palla, F. | title = The Formation of Stars | publisher = Wiley VCH | date = 2004 | doi = 10.1002/9783527618675 | isbn = 978-3-527-61867-5}}</ref> Luminosity functions are used to study the properties of large groups or classes of objects, such as the [[star]]s in [[star cluster|cluster]]s or the [[galaxy|galaxies]] in the [[Local Group]].

Note that the term "function" is slightly misleading, and the luminosity function might better be described as a luminosity ''distribution''. Given a luminosity as input, the luminosity function essentially returns the abundance of objects with that luminosity (specifically, number density per luminosity interval).

==Main sequence luminosity function==
The main sequence luminosity function maps the distribution of [[main sequence]] stars according to their luminosity. It is used to compare star formation and death rates, and [[stellar evolution|evolutionary models]], with observations. Main sequence luminosity functions vary depending on their host galaxy and on selection criteria for the stars, for example in the [[Solar neighbourhood]] or the [[Small Magellanic Cloud]].<ref>{{cite journal |bibcode=1977ApJ...216..372B |title=A main-sequence luminosity function for the Large Magellanic Cloud |last1=Butcher |first1=H. |journal=The Astrophysical Journal |year=1977 |volume=216 |page=372 |doi=10.1086/155477 |doi-access=free }}</ref>

==White dwarf luminosity function==
The '''white dwarf luminosity function''' ('''WDLF''') gives the number of [[white dwarf]] stars with a given luminosity. As this is determined by the rates at which these stars form and cool, it is of interest for the information it gives about the [[physics]] of white dwarf cooling and the age and history of the [[Milky Way|Galaxy]].<ref>{{cite journal |last1=Claver |first1=C. F. |last2=Winget |first2=D. E. |last3=Nather |first3=R. E. |author-link3=Ed Nather |last4=MacQueen |first4=P. J. |year=1998 |title=The Texas Deep Sky Survey: Spectroscopy of Cool Degenerate Stars |journal=American Astronomical Society Meeting Abstracts |volume=193 |bibcode=1998AAS...193.3702C}}</ref><ref name="cosmochronology">{{cite journal |bibcode=2001PASP..113..409F |title=The Potential of White Dwarf Cosmochronology |last1=Fontaine |first1=G. |last2=Brassard |first2=P. |last3=Bergeron |first3=P. |journal=Publications of the Astronomical Society of the Pacific |year=2001 |volume=113 |issue=782 |page=409 |doi=10.1086/319535 |s2cid=54970082 |doi-access=free }}</ref>

==Schechter luminosity function==
The '''Schechter luminosity function'''<ref>{{Cite journal |last=Schechter |first=P. |date=1976-01-01 |title=An analytic expression for the luminosity function for galaxies. |bibcode=1976ApJ...203..297S |journal=The Astrophysical Journal |volume=203 |pages=297–306 |doi=10.1086/154079 |issn=0004-637X}}</ref> <math>\phi</math> provides an approximation of the abundance of galaxies in a luminosity interval <math>[L+dL]</math>. The luminosity function has units of a number density <math>n</math> per unit luminosity and is given by a power law with an exponential cut-off at high luminosity
:<math>dn(L) = \phi~ dL = \phi^* \left(\frac{L}{L^*}\right)^\alpha \mathrm{e}^{-L/L^*} d\left(\frac{L}{L^*}\right),</math>
where <math>L^*</math> is a characteristic galaxy luminosity controlling the cut-off, and the normalization <math>\,\!\phi^*</math> has units of number density.

Equivalently, this equation can be expressed in terms of log-quantities<ref>{{Cite journal |last1=Sobral |first1=David |last2=Smail |first2=Ian |last3=Best |first3=Philip N. |last4=Geach |first4=James E. |last5=Matsuda |first5=Yuichi |last6=Stott |first6=John P. |last7=Cirasuolo |first7=Michele |last8=Kurk |first8=Jaron |date=2013-01-01 |title=A large Hα survey at z = 2.23, 1.47, 0.84 and 0.40: the 11 Gyr evolution of star-forming galaxies from HiZELS★ |journal=Monthly Notices of the Royal Astronomical Society |volume=428 |issue=2 |pages=1128–1146 |doi=10.1093/mnras/sts096 |issn=0035-8711|doi-access=free |arxiv=1202.3436 |bibcode=2013MNRAS.428.1128S }}</ref> with

:<math>dn(L) = \ln(10) \phi^* \left(\frac{L}{L^*}\right)^{\alpha+1} \mathrm{e}^{-L/L^*} d\left(\log_{10}L\right).</math>

The galaxy luminosity function may have different parameters for different populations and environments; it is not a universal function. One measurement from field galaxies is <math>\alpha=-1.25,\ \phi^* = 1.2 \times 10^{-2} \ h^3 \ \mathrm{Mpc}^{-3}</math>.<ref>{{cite book
|title=Galaxy Formation |last=Longair |first=Malcolm |date=1998 |publisher=[[Springer-Verlag]] |isbn=978-3-540-63785-1
}}</ref>

It is often more convenient to rewrite the Schechter function in terms of [[Magnitude (astronomy)|magnitudes]], rather than luminosities. In this case, the Schechter function becomes:

:<math> n(M)~ dM = (0.4 \ \ln 10) \ \phi^* \ [ 10^{ 0.4 ( M^* - M ) } ]^{ \alpha + 1} \exp [ -10^{ 0.4 ( M^* - M ) } ] ~ dM .
</math>

Note that because the magnitude system is logarithmic, the power law has logarithmic slope <math> \alpha + 1 </math>. This is why a Schechter function with <math> \alpha = -1 </math> is said to be flat.

Integrals of the Schechter function can be expressed via the [[incomplete gamma function]]
:<math> \int_a^b \left(\frac{L}{L^*}\right)^\alpha e^{-\left(\frac{L}{L^*}\right)} d \left(\frac{L}{L^*}\right)=\Gamma(\alpha+1,a)-\Gamma(\alpha+1,b) </math>

Historically, the Schechter luminosity function was inspired by the [[Press–Schechter formalism|Press–Schechter model]]. <ref name="Barkana">{{cite book |last1=Barkana |first1=Rennan |title=The Encyclopedia of Cosmology, Volume 1: Galaxy Formation and Evolution |date=2018 |volume=1 |publisher=World Scientific |doi=10.1142/9496 |isbn=9789814656221 |s2cid=259542973 |url=https://www.worldscientific.com/worldscibooks/10.1142/9496#t=aboutBook}}</ref> However, the connection between the two is not straight forward. If one assumes that every [[dark matter halo]] hosts one galaxy, then the Press-Schechter model yields a slope <math>\alpha\sim-3.5</math> for galaxies instead of the value given above which is closer to -1. The reason for this failure is that large halos tend to have a large host galaxy and many smaller satellites, and small halos may not host any galaxies with stars. See, e.g., [[halo occupation distribution]], for a more-detailed description of the halo-galaxy connection.

==References==
{{reflist}}

[[Category:Stellar astronomy]]
[[Category:Galaxies]]
[[Category:Photometry]]
[[Category:Equations of astronomy]]

Luminosity function (astronomy) - Revision history

2001:5A8:4613:AF00:6D58:CE16:7D8C:50CC: /* Schechter luminosity function */Fixed mistaken edit.