Form of the IMF Initial mass function
1 form of imf
1.1 salpeter (1955)
1.2 miller-scalo (1979)
1.3 chabrier (2003)
1.4 kroupa (2001)
1.5 uncertainties
form of imf
initial mass function
the imf stated in terms of series of power laws,
n
(
m
)
d
m
{\displaystyle n(m)\mathrm {d} m}
(sometimes represented
ξ
(
m
)
Δ
m
{\displaystyle \xi (m)\delta m}
), number of stars masses in range
m
{\displaystyle m}
m
+
d
m
{\displaystyle m+\mathrm {d} m}
within specified volume of space, proportional
m
−
α
{\displaystyle m^{-\alpha }}
,
α
{\displaystyle \alpha }
dimensionless exponent. imf can inferred present day stellar luminosity function using stellar mass-luminosity relation model of how star formation rate varies time. commonly used forms of imf kroupa (2001) broken power law , chabrier (2003) log-normal.
salpeter (1955)
the imf of stars more massive our sun first quantified edwin salpeter in 1955. work favoured exponent of
α
=
2.35
{\displaystyle \alpha =2.35}
. form of imf called salpeter function or salpeter imf. shows number of stars in each mass range decreases rapidly increasing mass. salpeter initial mass function is
ξ
(
m
)
Δ
m
=
ξ
0
(
m
m
s
u
n
)
−
2.35
(
Δ
m
m
s
u
n
)
.
{\displaystyle \xi (m)\delta m=\xi _{0}\left({\frac {m}{m_{\mathrm {sun} }}}\right)^{-2.35}\left({\frac {\delta m}{m_{\mathrm {sun} }}}\right).}
where
ξ
0
{\displaystyle \xi _{0}}
constant relating local stellar density.
miller-scalo (1979)
later authors extended work below 1 solar mass (m☉). glenn e. miller , john m. scalo suggested imf flattened (approached
α
=
1
{\displaystyle \alpha =1}
) below 1 solar mass.
chabrier (2003)
chabrier 2003 individual stars:
ξ
(
m
)
Δ
m
=
0.158
(
1
/
(
ln
(
10
)
m
)
)
exp
[
−
(
log
(
m
)
−
log
(
0.08
)
)
2
/
(
2
×
0.69
2
)
]
{\displaystyle \xi (m)\delta m=0.158(1/(\ln(10)m))\exp[-(\log(m)-\log(0.08))^{2}/(2\times 0.69^{2})]}
m
<
1
,
{\displaystyle m<1,}
ξ
(
m
)
=
k
m
−
α
{\displaystyle \xi (m)=km^{-\alpha }}
m
>
1
,
α
=
2.3
±
0.3
{\displaystyle m>1,\alpha =2.3\pm 0.3}
chabrier 2003 stellar systems (e.g. binaries):
ξ
(
m
)
Δ
m
=
0.086
(
1
/
(
ln
(
10
)
m
)
)
exp
[
−
(
log
(
m
)
−
log
(
0.22
)
)
2
/
(
2
×
0.57
2
)
]
{\displaystyle \xi (m)\delta m=0.086(1/(\ln(10)m))\exp[-(\log(m)-\log(0.22))^{2}/(2\times 0.57^{2})]}
m
<
1
,
{\displaystyle m<1,}
ξ
(
m
)
=
k
m
−
α
{\displaystyle \xi (m)=km^{-\alpha }}
m
>
1
,
α
=
2.3
±
0.3
{\displaystyle m>1,\alpha =2.3\pm 0.3}
kroupa (2001)
pavel kroupa kept
α
=
2.3
{\displaystyle \alpha =2.3}
above half solar mass, introduced
α
=
1.3
{\displaystyle \alpha =1.3}
between 0.08-0.5 m☉ ,
α
=
0.3
{\displaystyle \alpha =0.3}
below 0.08 m☉.
ξ
(
m
)
=
m
−
α
,
{\displaystyle \xi (m)=m^{-\alpha },}
α
=
0.3
{\displaystyle \alpha =0.3}
m
<
0.08
,
{\displaystyle m<0.08,}
α
=
1.3
{\displaystyle \alpha =1.3}
0.08
<
m
<
0.5
,
{\displaystyle 0.08<m<0.5,}
α
=
2.3
{\displaystyle \alpha =2.3}
m
>
0.5
{\displaystyle m>0.5}
uncertainties
there large uncertainties concerning substellar region. in particular, classical assumption of single imf covering whole substellar , stellar mass range being questioned in favour of two-component imf account possible different formation modes of substellar objects. i.e. 1 imf covering brown dwarfs , very-low-mass stars on 1 hand, , ranging higher-mass brown dwarfs massive stars on other. note leads overlap region between 0.05 , 0.2 m☉ both formation modes may account bodies in mass range.
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