1

Lecture 10: Chemical Evolution of Galaxies Metalicity evolution Z( t ) ( vs galaxy type )

Processes that alter the metalicity: 1.  Type-II SNe enrich the ISM. 2.  Low-mass stars form from enriched ISM

and “lock-up” metals.

3. Primordial gas falls in from IGM.

4. ISM ejected into IGM. (e.g. SN explosions, galaxy collisions)

Closed Box model: 1 and 2 only. Accreting Box: 1,2,3. Leaky Box: 1,2,4.

M0 = total mass

MG( t ) = mass of gas in ISM

MZ ( t ) = mass of metals in ISM

M* ( t ) = mass locked up in stars and remnants Mass conservation:

We also know:

To derive:

We will find:

M*(t) =M

0!M

G(t)

µ(t) "M

G(t)

M0

µ(0) =1

Z(t) "M

Z(t)

MG

(t)Z(0) = 0

Z(µ(t))

Metalicity Evolution: Z(t)

ISM Recycling Model

ISM

MG

(t), MZ(t)

Z !MZM

G

!

IGM

X = 0.75,Y = 0.25, Z = 0

M*(t)

lo-mass stars

+ NS, BH

dMG= !dM

*

dMZ = (y! Z ) dM*

dMG= dM

IN! dM

OUT

Yield: y = mass of metals returned to ISM per mass turned into low-mass stars and remnants

ISM Recycling Model

ISM

MG

(t), MZ(t)

Z !MZM

G

!

IGM

X = 0.75,Y = 0.25, Z = 0

M*(t)

lo-mass stars

+ NS, BH

dMG= !dM

S

dMZ= !Z dM

S

!

dMG

= 1"#( ) dMSdM

Z= Z

SN1"#( ) dMS

!

SN II

Z" ZSN

!

dM*

=" dMS

!

ZSN" 0.13

# " 0.93

From Salpeter IMF and SN1987A:

dMG= dM

IN! dM

OUT

The “Yield”Mass is conserved ( gas! stars )

dMG = " dM* = "! dMS

Metals are lost to stars, but enriched gas is returned by SNe:

dMZ = " Z dMS + ZSN 1"!( ) dMS

= "! Z +! Z( )" Z + ZSN 1"!( )#$ %&dM*

!

'

()

*

+,

=ZSN " Z( ) 1"!( )

!" Z

#

$-

%

&. dM/ 0 y" Z( ) dM*

Yield: y = ZSN " Z( )1"!

!

'

()

*

+,

Initial yield: y0 = ZSN1"!

!

'

()

*

+,= (0.13)

0.07

0.93= 0.01

!

ZSN" 0.13

# " 0.93

From Salpeter IMF and SN1987A:

Differentiate:

!

Z(t) "M

Z(t)

MG(t)

!Z = !M

Z

MG

!

"#

$

%&=

!MZ

MG

+MZ!

1

MG

!

"#

$

%&

=!M

Z

MG

+MZ'!M

G

MG

2

!

"#

$

%&

= 1

MG

!MZ'M

Z

MG

!MG

!

"#

$

%&

= 1

MG

Z ' y( ) !MG ' Z !MG( )

= 'y!M

G

MG

= 'y! ln(MG

)( )

Metalicity Evolution Z( t )

Definition of yield:

!MZ= y! Z( ) !M*= Z ! y( ) !MG

!

y =1/ x

!y

!x= !1/ x

2

!y = !!x / x2

2

Closed Box with constant Yield Integrate (with y = constant ) :

At Z = 0, MG = M0 :

Note that as µ => 0, Z => ∞

Impossible ! :-(

What went wrong? Yield is not quite constant.

Z = !y ln MG( )+C

0 = ! y ln M0( )+ C " C = y ln M0( )

! Z = " y lnM

G

M0

#

$%

&

'(= "y ln µ( )

!Z = !y!M

G

MG

!

Z

!

µ

!

1

y

y = ZSN ! Z( )1!!

!

"

#$

%

&' y0 = ZSN

1!!

!

"

#$

%

&'

"Z = !y " lnµ( ) = Z ! ZSN( )1!!

!

"

#$

%

&'" lnµ( )

!Z

Z ! ZSN=

1!"

"

"

#$

%

&'! lnµ( )

ln Z ! ZSN( ) =1!!

!

"

#$

%

&' ln µ( )+C

Z ! ZSN = A µ1!!

!

"

#$

%

&'

A = eC= !ZSN

Z = ZSN 1!µ1!!

!

"

#$

%

&'"

#$$

%

&'' y = y0 µ

1!!

!

"

#$

%

&'

Closed Box with varying Yield

!

Z

!

µ

!

1

y0!

ZSN

Yield is approx constant: (1-α) / α ~ 0.075, (0.1)0.075=0.98 But y => 0 and Z => ZSN as µ = > 0 ( from the last SN ).

!

µ

!

1

y0

y

Metalicity distribution of the Stars

!

Z!

µ

!

1

y0

!

ZSN

Metalicity of stars = Metalicity of gas from which they

formed.

“G dwarf problem”: very few sun-like stars (spectral type G) have metalicity below ½ solar. Closed Box Model FAILS: predicts that > ¼ of stars with Z < Z have Z < ½ Z""Why are there so few low-metalicity stars? What caused the rapid initial enrichment?

!

Z

M*(< Z )

M0

!

ZSNZ

What caused the initial enrichment?

ISM

MG

(t), MZ(t)

Z !MZM

G

!

IGM

X = 0.75,Y = 0.25, Z = 0

M*(t)

lo-mass stars

+ NS, BH

dMG= !dM

*

dMZ = (y! Z ) dM*

dMG= dM

IN! dM

OUT

dMOUT

= 0

IGM somehow enriched before galaxies form? First generation (Pop III) Z = 0 stars all high mass? Accreting Box model with low initial gas mass and Z=>y ?

Accreting Box varying Yield

y( Z(t) )!

Z(t)

!

µ(t) = gas mass total mass

!

S(t) = star mass total mass

Yield = y !Z

SN-Z( ) 1"!( )!

!

ZSN

= 0.13

!

" = 0.93

Assume star formation proportional to gas mass

( e.g. Elliptical galaxy)

Accrete Z = 0 gas, constant dMIN / dt , until t = 100.

Closed box for t > 100.

Result: Z( t ) rises until Z ~ y( Z( t ) )

M0(t) = !M t = total mass Accreting Box, constant gas mass

Approx: Z = -y ln(µ)

until Z = y. !

Z

!

µ

!

1

y

dMOUT = 0 dMIN = dM* => MG= constant dZ = dMZ / MG = ( y – Z ) dM* Z stops increasing when Z => y

ISM

MG

(t), MZ(t)

Z !MZM

G

!

IGM

X = 0.75,Y = 0.25, Z = 0

M*(t)

lo-mass stars

+ NS, BH

dMG= !dM

*

dMZ = (y! Z ) dM*

dMG= dM

IN! dM

OUT

3

Insert µ( t ) for each galaxy type into

Ellipticals:

Spirals:

Irregulars:

Z(t) = ! y ln e!t/t"( ) = y

t

t" for Z # y

Z(t) = y otherwise

!

µ(t) =1-" ˙ M t

M0

!

µ(t) = f"M t

M0

Z(t) = ! y ln 1! f! !M t

M0

"

#$

%

&' for Z ( y

Z(t) = y otherwise

Z(t) = ! y ln 1!! !M t

M0

"

#$

%

&' for Z ( y

Z(t) = y otherwise

!

µ(t) = e -t / t"( )

Z(t) = ! y ln µ(t)( )

for Z < y

!

Z

!

µ

!

1

y

!

Z

y

!

t

!

t"

!

µ

!

1

!

t

!

t"

!

µ(t) = e" t / t#

Z(t) = y t / t!

Ellipticals:

!

Z

y

!

t

!

t"

!

µ

!

1

!

t

!

t"

!

µ(t) =1" t / t#

Z(t) = ! y ln 1! t / t"( )

Spirals:

!

t" =M

0

# ˙ M

Z(t) = ! y ln µ(t)( )

for Z < y

!

µ(t) "1# t / t$

Z(t) ! "y ln 1" t / t#( )

!

Z

y

!

t

!

t"

!

µ

!

1

!

t

!

t"

Irregulars:

!

t" = fM

0

# ˙ M burst

f