Homework # 8 Solutions
1. Show that a simple closedbox model with instantaneous recycling results in
the gasphase metallicity increasing linearly with time.
If this is true, what
cosmic time does [Fe/H]=3 correspond to?
From the notes, we see that
Z
=
Z
0
+
pν
(1

R
)
t
, but only if the Schmidt
law with
N
= 1 is assumed. Since
Z
(
T

4
.
5 Gyr)) =
Z
, where
T
=today,
Z
(
T
) =
Z
+ (4
.
5 Gyr)
pν
(1

R
). If
Z
0
= 0,
Z
(
T
) =
Z T/
(
T

4
.
5 Gyr)
’
1
.
6
Z
. Then
t
= 7
.
5 Myr.
Also show in this model that the distribution of stars in metallicity
dN/d
ln
Z
has a maximum where
Z
=
p
, the effective yield of heavy elements defined in
the lecture notes.
Since
dN/d
ln
Z
=
M
Ze

(
Z

Z
0
)
/p
, where
N
is the number of stars in an
interval of
Z
, the maximum will be at
Z
=
p
.
2. Explore the evolution of the solar neighborhood with a galactic chemical evo
lution model with instaneous recycling and infall which obeys
f
(
t
) =
A
(
1

t/τ
f
)
.
Assume an infall timescale of
τ
f
= 24 Gyr and initial densities Σ
g
(0)+Σ
*
(0) =
Σ)
T
(0). Determine the value of
A
from the observed surface density Σ
T
(
T
) =
50
M
pc

2
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 Spring '11
 Lattimer
 Galaxies, new value, Gyr, σg, instantaneous recycling results

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