Download AST 105: Introduction to the Solar System HOMEWORK # 3

AST 105: Introduction to the Solar System
HOMEWORK # 3 SOLUTIONS
1. The Sun’s power output is 3.8 · 1026 watts. How much energy does the Sun
radiate in one second?
A watt is a unit of power, equal to 1 joule of energy radiated per second.
Therefore the amount of sunlight radiated in 1 second is 3.8 · 1026 joules.
2. Suppose a light wave has a wavelength of 1 meter. What frequency does it have?
For a wave, wavelength times frequency = speed of wave. The speed of
light is c = 3 · 108 m/s. So
wavelength =
3 · 108 m/s
speed
=
= 3 · 108 s−1 = 3 · 108 Hz = 300 MHz.
frequency
1 m
3. Does red light have a longer or shorter wavelength than blue light? Would a
photon of red light contain more or less energy than a photon of blue light?
Red light has a longer wavelength, a smaller frequency, and thus a photon
energy which is smaller than blue light.
4. In the movie Contact, Jodi Foster is shown listening on headphones to signals
from space. Why is this fantasy?
Human ears are not sensitive to radio waves. The waves have to be converted electronically to audible frequencies which is not a unique process. Audible frequencies could be encoded into radio waves, but how would we know
what the algorithm was? On Earth we use two main encoding systems, frequency modulation (FM) and amplitude modulation (AM) but many other
codes could be devised.
5. Would the photons of light leaving a hot star have, on average, a higher or lower
energy than those leaving a cooler star? Would a hotter star appear redder or
bluer than a colder star?
According to Wien’s Law, a hotter star with a higher temperature would
have a peak emission at higher frequencies and higher energies than a cooler
star, so the average photon emitted would have a higher energy. By the same
reasoning, a hotter star appears to us to be bluer.
6. How many protons and how many neutrons are contained in the average nucleus
of the oxygen atom?
Normal oxygen has 8 protons and 8 neutrons.
What is the atomic number of oxygen?
Oxygen has 8 protons and an atomic number of 8.
What is the average atomic weight of oxygen?
8 + 8 = 16
Is the average atomic weight of a molecule of CO2 heavier or lighter than
the average atomic weight of a molecule of N2 ?
Average atomic weights of C, N and O are 12, 14 and 16. CO2 weighs 44
units and N2 weighs 28 units so CO2 is heavier.
7. What is the weight of air above each square foot of the Earth’s surface?
Normal air pressure at sea level is 1 bar or 1 atmosphere which is about
14.7 pounds per square inch. A square foot has 144 square inches, so the weight
would be 14.7 × 144 = 2117 pounds.
8. If a cloud of cool gas exists between us an a bright continuous light source, what
kind of spectrum would we see?
An absoption spectrum superimposed on a continuous spectrum.
If we observed the same cloud without the continuous light source behind it,
what kind of spectrum would we see?
An emission spectrum.
9. Does a blue star emit more or less infrared energy per second than a red star,
assuming the two stars have the same diameters?
According to Wien’s Law, a blue star is hotter than a red star. Per unit
area, a hot object always emits more energy per second at every wavelength
than a cooler object, so indeed a blue star emits more infrared energy per
second than a red star.
10. Consider three stars: Star A has a radius of 1 R⊙ , the same radius as the Sun,
and a surface temperature of 6000 K. Star B has a radius of 2R⊙ and a surface
temperature of 12,000 K. Star C has a radius of 2R⊙ and a surface temperature
of 3000 K. What is the luminosity or power output of star B/star A?
The law of radiation from a spherical object is L = 4πR2σT 4 where L is
the luminosity, R is the radius, T is the temperature and σ is the radiation
constant. Therefore
2 4
LA
RA
TA
=
= 22 × 24 = 26 = 64
LB
RB
TB
What is the luminosity of star C/star A?
LA
=
LC
RA
RC
2 TA
TC
4
= 22 × (1/2) 24 = 2−2 = 1/4
What are the wavelengths of the maximum intensities of light from stars A, B
and C?
Wien’s Law says
2.9 · 106 nm
λmax =
;
T (K)
where a nm (nanometer) is 10−9 m, so
λmax,A =
2.9 · 107
nm = 483.3 nm;
6000
λmax,B =
2.9 · 107
nm = 241.6 nm;
12000
λmax,C =
2.9 · 107
nm = 966.6 nm.
3000
11. The wavelength of a normal red laser is 650 nm (at rest). If this laser was on
a spaceship moving toward you at 20% of the speed of light, what wavelength of
laser light would you measure and what color would it appear to be?
The Doppler effect says that if a source is approaching you, you would see
the wavelength get shorter. The percentage change of the wavelength is the
velocity of approach divided by the speed of light:
∆λ
v
= ,
λrest
c
where λrest = 650 nm is the wavelength of the laser at rest and v/c = −0.2
(the negative sign means the source is approaching you). So the wavelength of
light you would see is
v
λ = λrest + ∆λ = λrest 1 +
= 650 × (1 − 0.2) nm = 520 nm.
c
This wavelength is the same as the color of a green laser.