Download Astronomy 114 Problem Set # 5 Due: 04 Apr 2007 SOLUTIONS 1 1

Astronomy 114
Due:
Problem Set # 5
04 Apr 2007
SOLUTIONS
1 1 ton of TNT produces an explosion with an energy of 4.2 × 109 joules (1 ton =
907 kg). How much mass is converted into energy in such an explosion?
According to Einstein mass-energy equation: E = mc2 . The amount of mass
that must be converted to energy to produce as much energy as the TNT
explosion is found by solving the above equation for mass:
m=
E
4.2 × 109
=
kg = 4.67 × 10−8 kg
c2
(3 × 108 )2
2 If the entire mass of the TNT in the last problem were completely converted into
energy, how much energy would be released?
According to Einstein mass-energy equation, the released energy is:
E = mc2 = 907 × (3 × 108 )2 joules = 8.16 × 1019 joules.
Note that this is ten orders of magnitude larger than the chemical TNT explosion. This should help give you some feel for the power in fusion reactions.
3 The Sun has a luminosity of 3.83 × 1026 joules/sec. How much mass does the
Sun lose each second?
According to solar luminosity, each second the energy that the sun gives out
is 3.83 × 1026 joules, then from Einstein mass-energy equation, each second
the mass the Sun lose is:
m=
E
3.83 × 1026
=
kg = 4.26 × 109 kg
2
8
2
c
(3 × 10 )
Note that at the Sun’s main-sequence age (1010 years), the Sun will have lost:
4.26 × 109 kg/s × 1010 years364.25
hours
s
days
× 24
× 3600
= 1.3 × 1027 kg
year
day
day
which is 0.1% of the Suns’ total mass.
4 The central star in a newly formed planetary nebula has a luminosity of 1000
L⊙ and a surface temperature of 100,000 K. How big is the star?
The luminosity for a star is L=4πR2 σT 4 , thus we can get
R2 T 4
L
=
=
L⊙
R⊙ 2 T⊙ 4
so
R
=
R⊙
thus
v
u
u LT⊙ 4
t
L⊙ T 4
=
s
R
R⊙
!2
T
T⊙
!4
1000 × 58004
= 0.11
1000004
R = 0.11R⊙ = 7.66 × 107 m
5 The energy-generation rate in a star depends sensitively on the core temperature.
Use this fact to explain why a relation between a star’s mass and its luminosity
should exist, and why it is not surprising that L ∝ M 3.5 rather than just L ∝ M.
One’s first guess might be that the L ∝ M since a larger star will have proportionately more mass available for fusion reactions. However, the larger
stellar mass exerts a larger pressure on the star’s core and increases its temperature. Since the luminosity depends on the the energy-generation rate and
the energy generation rate increases rapidly with temperature, the total luminosity will increase more steeply than a simple linear proprotionality. For
example, if the core temperature were proportional to mass and the energy
generation rate were proportional to T 2 , we would estimate that L ∝ M 3 ,
one power of M for the larger overall mass available for fusion and M 2 for
the increased fusion rate due to temperature. A detailed calculation taking
the mass, pressure, and details of the nuclear energy generation rate into
account gives L ∝ M 3.5 .
6 How do astronomers know that the stars in globular clusters are old?
The age of the cluster can be determined by the turn-off point of the main
sequence on the HR diagram. Because:
(a) The main-sequence age is proportional to stellar mass
(b) The stellar mass is proportional to stellar luminosity
(c) It follows that the cluster contains no higher-mass main-sequence star
above this turn off.
Therefore, the main-sequence age of the turn-off then tells us the age of the
cluster. For globular cluster, this age is typically 1010 years; this is a large
fraction of the age of the Universe (as we will learn in a few weeks).