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Problem 5.142
Explorer 7 was launched on October 13, 1959, with an apogee altitude above the Earth’s surface of
1073 km and a perigee altitude of 573 km above the Earth’s surface. Its orbital period was 101:4 min.
Using this information, calculate Gmefor the Earth and compare it with gr2
e.
Photo credit: NASA
Solution
Recalling that the radius of the Earth
reD6371 km
, we begin with computing the orbit’s radii at perigee and
apogee. Specifically, we have
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Dynamics 2e 1071
Problem 5.143
Explorer 7 was launched on October 13, 1959, with an apogee altitude above the Earth’s surface of
1073 km and a perigee altitude of 573 km above the Earth’s surface. Its orbital period was 101:4 min.
Determine the eccentricity of the Explorer 7’s orbit, as well as its speeds at perigee and apogee.
Photo credit: NASA
Solution
Recalling that the radius of the Earth
reD6371 km
, we begin with computing the orbit’s radii at perigee and
apogee. Specifically, we have
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permission of McGraw-Hill, is prohibited.
1072 Solutions Manual
gD9:81 m=s2
is the acceleration due to gravity on the surface of the Earth, for the speeds at perigee and
apogee we have
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Dynamics 2e 1073
Problem 5.144
Ageosynchronous equatorial orbit is a circular orbit above the Earth’s
equator that has a period of
1
day (these are sometimes called geostationary
orbits). These geostationary orbits are of great importance for telecommu-
nications satellites because a satellite orbiting with the same angular rate
as the rotation rate of the Earth will appear to hover in the same point in
the sky as seen by a person standing on the surface of the Earth. Using
this information, determine the altitude
hg
and radius
rg
of a geostationary
orbit (in miles). In addition, determine the speed
vg
of a satellite in such an
orbit (in miles per hour).
gr2
e)rgD ⌧2
4⇡2!1=3
To find hgobserve that
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Problem 5.145
An artificial satellite is launched from an altitude of
500 km
with
a velocity
vP
that is parallel to the surface of the Earth. Requiring
that the altitude at apogee be
20;000 km
, determine the velocity
at
B
, that is, the position in the orbit when the velocity is first
orthogonal to the launch velocity.
Solution
The radius of Earth is reD6:371⇥106m so
rPDreC500 km )rPD6:871⇥106m;
rADreC20;000 km )rAD26:37⇥106m;
aD1
2.rACrP/)aD16:62⇥106m;
bDprArP)bD13:46⇥106m:
The distance from the center of Earth to Bis
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Dynamics 2e 1075
Problem 5.146
The mass of the planet Jupiter is 318 times that of Earth, and its equatorial radius is
71;500 km
. If a space
probe is in a circular orbit about Jupiter at the altitude of the Galilean moon Callisto (orbital altitude
1:812⇥106km
), determine the change in speed
v
needed in the outer orbit so that the probe reaches a
minimum altitude at the orbital radius of the Galilean moon Io (orbital altitude
3:502⇥105km
). Assume
that the probe is at the maximum altitude in the transfer orbit when the change in speed occurs and that
change in speed is impulsive, that is, it occurs instantaneously.
Solution
Let
rc
and
ri
denote the radii of Callisto’s and Io’s orbits, respectively. The radius
rc
is also the radius of the
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where
me
is the mass of the Earth. Hence, applying Eq. (4), we can determine the speeds
v1
and
v2
as
follows:
e
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Dynamics 2e 1077
Problem 5.147
The on-orbit assembly of the International Space Station (ISS) began
in 1998 and continues today. The ISS has an apogee altitude above
the Earth’s surface of
341:9 km
and a perigee altitude of
331:0 km
above the Earth’s surface. Determine its maximum and minimum
speeds in orbit, its orbital eccentricity, and its orbital period. Re-
search its actual orbital period and compare it with your calculated
value.
Photo credit: NASA
aD1
2.rACrP/D6:707⇥106m:
Hence, using the (full precision value of the) above results, the maximum orbital speed is at perigee and is
given by
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Problem 5.148
The optimal way (from an energy standpoint) to transfer from one
circular orbit about a primary body
B
to another circular orbit is via
the so-called Hohmann transfer, which involves transferring from one
circular orbit to another using an elliptical orbit that is tangent to both
at the periapsis and apoapsis of the ellipse. The ellipse is uniquely
defined because we know
rP
(the radius of the inner circular orbit) and
rA
(the radius of the outer circular orbit), and therefore we know the
semimajor axis
a
by Eq. (5.88) and the eccentricity
e
by Eq. (5.87) or
Eqs. (5.90). Performing a Hohmann transfer requires two maneuvers,
the first to leave the inner (outer) circular orbit and enter the transfer
ellipse and the second to leave the transfer ellipse and enter the outer
(inner) circular orbit.
A spacecraft S1needs to transfer from circular low Earth parking
orbit with altitude
120 mi
above the surface of the Earth to a circular
geosynchronous orbit with altitude
22;240 mi
. Determine the change
in speed
vP
required at perigee
P
of the elliptical transfer orbit and
the change in speed
vA
required at apogee
A
. In addition, compute
the time required for the orbital transfer. Assume that the changes in
speed are impulsive, that is, they occur instantaneously.
Solution
Recalling that the radius of the Earth is reD3959 mi D3959.5280/ ft, the radii at perigee and apogee are
Let
v1
be the speed when the spacecraft is in a circular orbit with radius
rP
,
v2
be the speed at perigee for the
transfer orbit,
v3
be the speed at apogee for the transfer orbit, and
v4
be the speed at the circular orbit with
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
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where, in addition to using the (full precision) values of
rP
,
rA
, and
a
indicated in Eqs. (1)–(3), we have
used
gD32:2 ft=s2
and
reD3959.5280/ ft
. Thus, using the full precision values of
v1
,
v2
,
v3
, and
v4
, and
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