ASTRONOMY 294Z: The History of the Universe
Professor Barbara Ryden
SOLUTIONS TO PROBLEM SET # 4
1) [20 points]
Today, the average density of matter in the universe is
ρ
=
3
×
10

27
kg
/
m
3
.
If the matter consisted entirely of hydrogen atoms, how
many hydrogen atoms (on average) would be contained in a cubic meter of
the universe? If the matter consisted entirely of regulation baseballs (of mass
M
= 0
.
145 kg
apiece), how many baseballs (on average) would be contained
in a cubic astronomical unit of the universe?
The mass of a single hydrogen atom is
m
= 1
.
7
×
10

27
kg. In order for
the mass density to be
ρ
= 3
×
10

27
kg
/
m
3
, the number of hydrogen atoms
per cubic meter would have to be
n
=
ρ
m
=
3
×
10

27
kg
/
m
3
1
.
7
×
10

27
kg
= 1
.
76
/
m
3
.
(1)
Fewer than two atoms, on average, per cubic meter of space – that’s a very
low density.
The mass of a single baseball is
M
= 0
.
145 kg.
In order for the mass
density to be
ρ
= 3
×
10

27
kg
/
m
3
, the number of baseballs per cubic AU
would have to be
n
=
ρ
M
=
ˆ
3
×
10

27
kg
/
m
3
0
.
145 kg
!ˆ
1
.
5
×
10
11
m
1 AU
!
3
= 69
,
800
,
000
/
AU
3
.
(2)
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View Full Document2) [20 points]
The temperature of the cosmic background light today is
T
≈
3 K
.
At the time the universe became transparent, the temperature of the
cosmic background light was
T
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 Winter '08
 RYDEN
 Astronomy, Atom, Deuterium

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