Luminosity function (astronomy)

In astronomy, the luminosity function gives the number of stars or galaxies within a given volume of space as a function of their luminosity. It is a fundamental tool for understanding the distribution of intrinsic brightnesses of astronomical objects and for modeling the formation and evolution of stellar populations and galaxies.

The luminosity function is typically represented as Φ(L), where Φ(L) dL gives the number density of objects (number per unit volume) with luminosities between L and L + dL. It can also be expressed in terms of absolute magnitude, M, instead of luminosity, L. Since absolute magnitude is a logarithmic scale, the shape of the luminosity function changes when expressed in terms of magnitude, denoted as Φ(M). The relationship between luminosity and absolute magnitude is:

M = -2.5 log₁₀(L/L₀)

where L₀ is a reference luminosity.

Several mathematical forms are commonly used to approximate observed luminosity functions. Some of the most prevalent include:

• Schechter Function: This is a commonly used empirical model for describing the luminosity function of galaxies. It has the form:

  Φ(L) = (Φ*/L*) (L/L*)^α exp(-L/L*)

  where:

  • Φ* is a normalization constant (density).
  • L* is a characteristic luminosity where the power-law cuts off.
  • α is the faint-end slope, describing the behavior of the luminosity function at low luminosities.

  The Schechter function can also be expressed in terms of absolute magnitude:

  Φ(M) = (0.4 ln(10) Φ*) 10^{0.4(M* - M)(α + 1)} exp(-10^{0.4(M* - M)})

  where M* is the characteristic absolute magnitude corresponding to L*.

• Power Law: At sufficiently high luminosities, the Schechter function often resembles a power law. Simple power law functions are sometimes used to model specific portions of luminosity functions, especially in cases where limited data prevent a more complex fit.

The luminosity function is sensitive to various factors, including:

• Selection effects: Observational biases can significantly affect the observed luminosity function. Fainter objects are harder to detect at greater distances, leading to an underestimation of their number density, especially at the faint end of the luminosity function. This needs to be corrected for when estimating the true luminosity function.
• Cosmological parameters: The assumed cosmological model affects the derived distances to astronomical objects, which in turn affects the estimated luminosities and the derived luminosity function.
• Environment: The environment in which galaxies or stars reside (e.g., galaxy clusters vs. the field) can influence their formation and evolution, leading to different luminosity functions in different environments.

The luminosity function has applications in various areas of astronomy and cosmology, including:

• Estimating the total luminosity density of the Universe: By integrating the luminosity function, astronomers can estimate the total light emitted by all stars or galaxies in a given volume.
• Constraining models of galaxy formation and evolution: The shape and evolution of the luminosity function provide crucial constraints on models of how galaxies form and evolve over cosmic time.
• Determining distances to galaxies: By comparing the observed luminosity function of a distant galaxy cluster to a local luminosity function, astronomers can estimate the distance to the cluster using a method known as luminosity function distance.
• Studying the properties of stellar populations: The luminosity function of a star cluster or galaxy can be used to infer the age, metallicity, and star formation history of the stellar population.