In
astronomy
Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...
, the initial mass function (IMF) is an
empirical
Empirical evidence for a proposition is evidence, i.e. what supports or counters this proposition, that is constituted by or accessible to sense experience or experimental procedure. Empirical evidence is of central importance to the sciences and ...
function that describes the initial
distribution Distribution may refer to:
Mathematics
*Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations
* Probability distribution, the probability of a particular value or value range of a vari ...
of masses for a population of stars. The IMF is an output of the process of
. The IMF is often given as a
probability distribution function Probability distribution function may refer to:
* Probability distribution
* Cumulative distribution function
* Probability mass function
* Probability density function
In probability theory, a probability density function (PDF), or density ...
(PDF) for the mass at which a star enters the
main sequence
In astronomy, the main sequence is a continuous and distinctive band of stars that appears on plots of stellar color versus brightness. These color-magnitude plots are known as Hertzsprung–Russell diagrams after their co-developers, Ejnar Her ...
(begins
hydrogen fusion
Stellar nucleosynthesis is the creation (nucleosynthesis) of chemical elements by nuclear fusion reactions within stars. Stellar nucleosynthesis has occurred since the original creation of hydrogen, helium and lithium during the Big Bang. As a ...
). The distribution function can then be used to construct the
mass distribution
In physics and mechanics, mass distribution is the spatial distribution of mass within a solid body. In principle, it is relevant also for gases or liquids, but on Earth their mass distribution is almost homogeneous.
Astronomy
In astronomy mass d ...
(the
histogram
A histogram is an approximate representation of the distribution of numerical data. The term was first introduced by Karl Pearson. To construct a histogram, the first step is to " bin" (or "bucket") the range of values—that is, divide the ent ...
of stellar masses) of a population of stars. It differs from the ''present day mass function'' (PDMF), the current distribution of masses of stars, due to the evolution and death of stars which occurs at different rates for different masses as well as dynamical mixing in some populations.
The properties and evolution of a star are closely related to its mass, so the IMF is an important diagnostic tool for astronomers studying large quantities of stars. For example, the initial mass of a star is the primary factor determining its
colour
Color (American English) or colour (British English) is the visual perceptual property deriving from the spectrum of light interacting with the photoreceptor cells of the eyes. Color categories and physical specifications of color are associ ...
,
luminosity
Luminosity is an absolute measure of radiated electromagnetic power (light), the radiant power emitted by a light-emitting object over time. In astronomy, luminosity is the total amount of electromagnetic energy emitted per unit of time by a st ...
, and lifetime. At low masses, the IMF sets the
Milky Way Galaxy
The Milky Way is the galaxy that includes our Solar System, with the name describing the galaxy's appearance from Earth: a hazy band of light seen in the night sky formed from stars that cannot be individually distinguished by the naked eye. ...
mass budget and the number of substellar objects that form. At intermediate masses, the IMF controls chemical enrichment of the
interstellar medium
In astronomy, the interstellar medium is the matter and radiation that exist in the space between the star systems in a galaxy. This matter includes gas in ionic, atomic, and molecular form, as well as dust and cosmic rays. It fills interstella ...
. At high masses, the IMF sets the number of core collapse
supernovae
A supernova is a powerful and luminous explosion of a star. It has the plural form supernovae or supernovas, and is abbreviated SN or SNe. This transient astronomical event occurs during the last evolutionary stages of a massive star or when a ...
that occur and therefore the kinetic energy feedback.
The IMF is relatively invariant from one group of stars to another, though some observations suggest that the IMF is different in different environments, and potentially dramatically different in early galaxies.
Form of the IMF
![Plot of various initial mass functions](https://upload.wikimedia.org/wikipedia/commons/d/d5/Plot_of_various_initial_mass_functions.svg)
The IMF is often stated in terms of a series of
power law
In statistics, a power law is a Function (mathematics), functional relationship between two quantities, where a Relative change and difference, relative change in one quantity results in a proportional relative change in the other quantity, inde ...
s, where
(sometimes also represented as
), the number of stars with masses in the range
to
within a specified volume of space, is proportional to
, where
is a dimensionless exponent. The IMF can be inferred from the present day stellar
luminosity function
A luminous efficiency function or luminosity function represents the average spectral sensitivity of human visual perception of light. It is based on subjective judgements of which of a pair of different-colored lights is brighter, to descri ...
by using the stellar
mass–luminosity relation In astrophysics, the mass–luminosity relation is an equation giving the relationship between a star's mass and its luminosity, first noted by Jakob Karl Ernst Halm. The relationship is represented by the equation:
:\frac = \left(\frac\right)^a
whe ...
together with a model of how the star formation rate varies with time. Commonly used forms of the IMF are the Kroupa (2001)
broken power law
In statistics, a power law is a functional relationship between two quantities, where a relative change in one quantity results in a proportional relative change in the other quantity, independent of the initial size of those quantities: one qua ...
and the Chabrier (2003) log-normal.
Salpeter (1955)
The IMF of stars more massive than our sun was first quantified by
Edwin Salpeter
Edwin Ernest Salpeter (3 December 1924 – 26 November 2008,) was an Austrian–Australian–American astrophysicist.
Life
Born in Vienna to a Jewish family, Salpeter emigrated from Austria to Australia while in his teens to escape the Nazis. He ...
in 1955. His work favoured an exponent of
. This form of the IMF is called the Salpeter function or a Salpeter IMF. It shows that the number of stars in each mass range decreases rapidly with increasing mass. The Salpeter Initial Mass Function is
:
where
is the
solar mass
The solar mass () is a standard unit of mass in astronomy, equal to approximately . It is often used to indicate the masses of other stars, as well as stellar clusters, nebulae, galaxies and black holes. It is approximately equal to the mass ...
, and
is a constant relating to the local stellar density.
Miller-Scalo (1979)
Later authors extended the work below one
solar mass
The solar mass () is a standard unit of mass in astronomy, equal to approximately . It is often used to indicate the masses of other stars, as well as stellar clusters, nebulae, galaxies and black holes. It is approximately equal to the mass ...
(). Glenn E. Miller and
John M. Scalo suggested that the IMF "flattened" (approached
) below one solar mass.
Kroupa (2001)
Pavel Kroupa
Pavel Kroupa (born 24 September 1963 in Jindřichův Hradec, Czechoslovakia) is a Czech-Australian astrophysicist and professor at the University of Bonn.
Biography and career
After the 1968 failure of Prague spring, Kroupa's family fled from ...
kept
above half a solar mass, but introduced
between and
below .
:
:
for
:
for