Astronomy 1 - Fall 1998
- Assignment #4 - Solution
by Larry Bogan October 7, 1998
Chaisson Chapter 2
Problems #7, #8, #10
Chiasson Chapter 3
Problems #1, #2, #3
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2-7 Jupiter's Moon Callisto orbits the planet at a
distance of 1.88 million km. Callisto's orbital period
about Jupiter is 16.7 days. What is the mass of
Jupiter?
Kepler's Third law as modified by Newton's Laws will be useful here:
P2 = a3/M
Where P(years), a(AU), M(Solar Masses)
By using the P and a of Callisto we can determine the mass
of Jupiter in Solar Masses.
P = 16.7 days/(365.25 days/year) = 0.0457 years
Gm = gigameter = million km.
a = 1.88 Gm/150 Gm/AU = 0.0125 AU
M = a3/P2 = (0.0125)3/(0.0457)2
M(Jupiter) = 0.000935 = 1/1070 mass of Sun
= 2x1030 kg/1079 = 1.87x1027 kg
NOTE: You might also use the relationship M = 4(pi)2a3/(GP2)
and use P and a in metric units to get M in kg. (same result)
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2-8 The acceleration of gravity at Earth's surface is 9.8
m/s2. What is the acceleration at altitudes of
(a) 100 km
(b) 1000 km
(c) 10,000 km
The acceleration of gravity is due to the attraction of an object
to the center of mass of the Earth. The force is
F = G M m/r2
The acceleration of the object is
a = F/m = GM/r2
At the surface r = RE = 6371 km and a = g = 9.8 m/s2
So using the proportionality
a = g (RE/r)2
(a) alt = 100 km then r = 6371 + 100 = 6471 km
a = 9.8 (6371/6471)2 = 9.5 m/s2
(b) alt = 1000 km then r = 6371 + 1000 = 7371 km
a = 9.8 (6371/7371)2 = 7.3 m/s2
(c) alt = 10,000 km them r = 16371 km
a = 9.8 (6371/16371)2 = 1.48 m/s2
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2-10 The Moon's mass is 5.97x1024 kg and its radius
is 1700 km. What is the speed of a spacecraft moving
in a circular orbit just above the lunar surface? What
is the escape speed from the Moon's surface?
CORRECTIONS: The mass of the Moon is 1/81 that of Earth
( = 5.97x1024kg) sooo.....
mass of the Moon is 5.97x1024 kg/81
Mmoon= 7.37x1022 kg
If you solved the problem with the given value
-OK if you got 11.2 km/s
-- Another error in the text book. The velocity of
an object in a circular orbit about a mass, M
is sqrt[GM/r] NOT sqrt[2GM/r]!
vorbit = sqrt[(6.67x10-11 N m2/kg2)(7.37x1022kg)/(1.70x1024kg)]
= 1700 m/s = 1.7 km/s
vescape = sqrt[2] vorbit = 1.414 (1.7 km/s) = 2.4 km/s
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1-1 A sound wave moving through water has a
frequency of 256 Hz and a wavelength of 5.7 m. what
is the speed of sound in water?
v = wave speed = frequency x wavelength
v = 256 s-1 5.7 m = 1459 m/s
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1-2 What is the wavelength of a 100 MHz radio
signal? Megahertz = 1 million Hertz
speed of light = 3x108 m/s
wavelength = (speed of light)/frequency
f = 100 MHz = 100,000,000 Hz
wavelength = 3x108/108 = 3 m
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1-3 What would be the frequency of an
electromagnetic wave having a wavelength equal to
the Earth's diameter? In what part of the
electromagnetic spectrum would such a wave lie?
diameter of the earth = 2 x 6371 km = 12,740,000 m
frequency = (3x108m/s)/(1.27x107m) = 23.6 Hz
- this is the very low frequency radio region.
(ridiculously low!!!)