Initial mass function (IMF)

The initial mass function specifies the distribution of stellar masses at birth. Writing

ξ (M) = d N / d M

means that

ξ (M) d M

counts the expected number born in a small mass interval. Counts per logarithmic interval differ:

d N / d ln M = M ξ (M)

. Confusing these conventions changes the reported slope even for the same population.

For a chosen finite birth-mass range,

\int ξ (M) d M

gives the number of stars and

\int M ξ (M) d M

their total mass. The mass and luminosity weighting of a stellar initial mass function explains why a small number of luminous massive stars can dominate the light without dominating the count. A present-day distribution also reflects deaths, continued star formation, unresolved companions and observational selection, so it is not automatically the birth distribution.

Table of contents
- Mass and luminosity weighting of a stellar initial mass function Initial mass function
- Power-law initial-mass-function tail coordinate Initial mass function

Mass and luminosity weighting of a stellar initial mass function ( $d M / d ln m \propto m^{2 - s}, d L / d ln m \propto m^{q + 1 - s}$ )

 0  0

If birth counts obey

d N / d m \propto m^{- s}

and individual luminosity scales as

m^{q}

, multiply by mass or luminosity and by

d m = m d ln m

to obtain the displayed logarithmic weights. With

s = 2.35

and

q = 4

, birth mass has weight

m^{- 0.35}

while light has weight

m^{2.65}

. Finite mass cutoffs are essential; stellar lifetimes also alter the luminosity weighting of an evolved population.

Power-law initial-mass-function tail coordinate ( $X (M) = (M_{m i n} / M)^{p - 1}$ )

 0  0

For an initial mass function proportional to

M^{- p}

M \geq M_{m i n}

with

p > 1

, integration and normalization give the displayed upper-tail fraction. The probability integral transform makes

X

uniform on

[0, 1]

for a randomly sampled birth mass. Evolutionary lifetime regions can then be measured as ordinary areas when the independent age coordinate is also uniform.

 Ancestors (6)

 Incoming links (43)

 View article source

 Discussion (0) New discussion

There are no discussions about this article yet.

 Articles by others on the same topic (1)

Initial mass function by

Wikipedia Bot  1

 View more

The Initial Mass Function (IMF) is a crucial concept in astrophysics that describes the distribution of masses for a population of stars when they form. It provides a statistical representation of how many stars are born within a certain mass range in a stellar population, essentially outlining the relationship between the number of stars and their masses at the time of formation.

 Read the full article

  See all articles in the same topic Create my own version