Lecture 11: Closed Box and Beyond

!

ISMMG (t), MZ (t)Z " MZ MG

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IGMX = 0.75,Y = 0.25, Z = 0

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M*(t)lo - mass stars + NS, BH!

dMG = "dMS

dMZ = "Z dMS

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dMG = 1"#( ) dMS

dMZ = ZSN 1"#( ) dMS

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SN IIZ" ZSN

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dM* =" dMS

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ZSN " 0.13# " 0.93

From Salpeter IMF and SN1987A:

The Closed Box Model “Closed Box” model with constant Yield:

Closed Box model ignores:

1.  IGM--ISM exchanges: IGM falls in, ISM blown out of galaxy

2. SN Ia, stellar winds, PNe, novae, etc.

3.  Initial enrichment by e.g. Pop.III stars prior to galaxy formation?

4.  Faster enrichment (more SNe) in denser regions of galaxy.

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Z(t) = " # ln µ(t)( )Metalicity Yield Gas fraction

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Z(t) "# as µ " 0

!

Yield = " #mass of new metals added to ISM by SNemass of ISM converted to long - lived stars

=ZSN - Z( ) 1$%( )

%

But (with infall):

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µ

!

1!

"

!

Z "min #,$# log(µ)[ ]

!

Z

!

µ

!

1!

"

!

Z

!

"

!

t

!

t"

!

µ

!

1

!

t

!

t"!

µ(t) = e" t / t#

!

Z(t) = " t / t#

Ellipticals:

!

Z

!

"

!

t

!

t"

!

µ

!

1

!

t

!

t"!

µ(t) =1" t / t#

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Z(t) = " # ln 1" t / t$( )

Spirals:

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t" =M0

# ˙ M !

Z(t) = " # ln µ(t)( )

The G-dwarf problem :Expected number of metal-poor stars in SN

Mass of stars with Z=0.25Z. compared to the mass in stars with the current metallicity of gas:

Expect ~50% of stars near Sun to have Z<0.25Z.

However only ~2% of F-G dwarf stars in SN have Z<0.25Z.

G-dwarf problem

M*(< 0.25Z.)M*(< 0.7Z.)

=[1! e!0.25Z ./! ][1! e!0.7Z ./! ]

~ 0.54

G-dwarf problem: Possible solutions 1.  Super-enriched local ISM (i.e., statistical fluke) 2.  System started with higher metallicity Zo~0.15Z. 3.  Close box not valid, i.e.,

1.  Gas lost through winds, i.e., leaky-box model 2.  Gas accreted, i.e., accreting-box model

Fraternali et al (2007)

Leaky Box model Winds of massive stars (AGB) and SN can eject gaseous material out of a galaxy. As this gas is heavily enriched it can lead to the loss of metals from the ISM

The leaky-box model Let us assume an outflow of processed material give by g(t), then: Conservation of mass gives us: Lets adopt an outflow rate proportional to the star-formation rate: c = arbitrary constant and dM*/dt is the star-formation rate. So total change in stellar mass can be written as:

!MZ = (Z " !)!M* " Zg

!Mg +!M* + g = 0

g = c!M*

!M* = !1

(1+ c)!MG

The leaky-box model Can now derive the metallicity evolution, as before: Now substitute for dMG from previous slide: i.e., same as before, outflow simply reduces the yield from to an effective yield. Not really surprising that a leaky-box lowers final metallicity and exacerbates the G-dwarf problem.

Z(t) ! MZ

MG

!Z = ! MZ

MG

"

#$

%

&'= ("

!MG

MG

!Z = ! "(1+ c)

!MG

MG

Accreting/Infall Model Galaxies continually accrete primordial gas.

The accreting-box model Lets assume primordial gas is accreted at the star-formation rate, i.e., gas replacement. As for the closed box model the production of metals per unit gas mass is the same, i.e., However the conservation of mass is now: In fact lets consider a replenishment model such that the gas mass stays constant:

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

!MG = !!M* + f (t)

!Z = 1MG

[(" ! Z )!M* ! Z!MG ]=1MG

(" ! Z )!M*

closed box (as before) accreting box

Integrating and assuming Zo=0: Assuming SF and infall have been progressing continuously i.e.,M*>>MG then Z~ρ And mass in stars which are more metal-poor than Z is: For the solar neighbourhood: MG~10M . /pc2 and M*~40 . /pc2 and for Z~ρ~0.71Z . Fraction of stars more metal-poor than 0.25 Z . Is M(<0.25)/M(<0.7) ~ 10% Closer to that observed than in the closed box model.

The accreting-box model

Z = ![1! e!M* /MG ]

M*(< Z ) = !MG ln(1! Z / !)

The G-dwarf problem

At least two possible solutions: 1)  Disc formed from pre-

enriched gas (left over from Bulge?).

2)  Significant gas infall has occurred so that the remaining gas is NOT indicative of the amount of star-formation which has occurred.