An Etymological Dictionary of Astronomy and Astrophysics
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An empirical power-law relation representing the number of clusters as a function of their mass. It is defined as: N(M)dM ∝ M ^-αdM, where the exponent α has an estimated value of about 2 and dM is the mass interval. It is believed that this is a universal law applying to a variety of objects including globular clusters, massive young clusters, and H II regions.

→ cluster; → mass; → function.

The mass distribution of → pre-stellar cores in → star-forming regions. The CMF is usually represented by dN/dM = M^α, where dM is the mass interval, dN the number of cores in that interval, and α takes different values in different mass ranges. In the case of → low-mass stars, it is found that the CMF resembles the → Salpeter function, although deriving the masses and radii of pre-stellar cores is not straightforward. The observational similarity between the CMF and the → initial mass function (IMF) was first put forth by Motte et al. (1988, A&A, 336, 150), and since then many other samples of dense cores have been presented in this context. For example, Nutter & Ward-Thompson (2007, MNRAS 374, 1413), using SCUBA archive data of the Orion star-forming regions, showed that the CMF can be fitted to a three-part → power law consistent with the form of the stellar IMF. Recent results, obtained using observations by the → Herschel Satellite, confirm the similarity between the CMF and IMF with better statistics (Könyves et al. 2010, A&A, 518, L106; André et al. 2010, A&A, 518, L102). Moreover, these works show that the CMF has a → lognormal distribution (i.e. dN/dlog M follows a → Gaussian form against log M), as is the case for the IMF at low masses (below about 1 solar mass).

→ core; → mass; → function.

→ core mass function.

→ dense; → core; → mass; → function.

A mathematical expression describing the relative number of stars found in different ranges of mass for a cluster of stars at the time of its formation. It is defined as φ(log M) = dN / dlog M ∝ M ^-Γ, where M is the mass of a star and N is the number of stars in a logarithmic mass interval. The value of the slope found by Salpeter (1955) for → low-mass and → intermediate-mass stars in the → solar neighborhood is Γ = 1.35. The IMF can be expressed also in linear mass units: χ(M) = dN / DM ∝ M ^-α. Note that χ(M) = (1 / M lm 10) φ(log M), and α = Γ + 1. In this formalism the Salpeter slope is α = 2.35. There is a third way for representing the IMF, in which the exponent is x = -α. The IMF is not a single power law over all masses, from → brown dwarfs to → very massive stars (Kroupa, 2002, Science 295, 82). Different slopes have been found for different mass segments, as follows: α = 1.3 for 0.08 ≤ M_solar < 0.5; α = 2.3 for 0.5 ≤ M_solar < 1; α = 2.3 for 1 ≤ M_solar. The IMF at low masses can be fitted by a → lognormal distribution (See Bastian et al., 2010, ARAA 48, 339 and references therein). See also → canonical IMF.

→ initial; → mass; → function.

1) The number of a class of objects as a function of their mass. → initial mass function (IMF); → present-day mass function (PDMF).
2) A numerical relation between the masses of the two components of a → spectroscopic binary that provides information on the relative masses of the two stars when the spectral lines of only one component can be seen. If M_p is the mass of primary (whose spectrum is known), M_s is the mass of secondary, and i the → angle of inclination of the orbit, the mass function is given by: (M_s³. sin³i) / (M_p + M_s)².

→ mass; → function.

The present number of stars on the → main sequence per unit logarithmic mass interval per square parsec. The PDMF is the basis for deriving the → initial mass function (IMF). This mass function is not corrected for stellar evolution nor losses through stellar deaths.

→ present; → day; → mass; → function.