PROBLEM 12.101
It was observed that as the Voyager I spacecraft reached the point of its trajectory closest to the planet Saturn,
it was at a distance of 3
185 10km from the center of the planet and had a velocity of 21.0 km/s. Knowing
that Tethys, one of Saturn’s moons, describes a circular orbit of radius 3
295 10km at a speed of 11.35 km/s,
determine the eccentricity of the trajectory of Voyager I on its approach to Saturn.
PROBLEM 12.102
A satellite describes an elliptic orbit about a planet of mass M.
Denoting by 0
r and 1,r respectively, the minimum and maximum
values of the distance r from the satellite to the center of the planet,
derive the relation
2
01
112GM
rr h
where h is the angular momentum per unit mass of the satellite.
PROBLEM 12.103
A space probe is describing a circular orbit about a planet of radius
R
. The altitude of the probe above the
surface of the planet is
R
and its speed is
v
0
. To place the probe in an elliptic orbit which will bring it closer
to the planet, its speed is reduced from
v
0
to
0
,v
where 1,
by firing its engine for a short interval of
time. Determine the smallest permissible value of
if the probe is not to crash on the surface of the planet.
PROBLEM 12.104
A satellite describes a circular orbit at an altitude of 19 110 km above
the surface of the earth. Determine (a) the increase in speed required at
point A for the satellite to achieve the escape velocity and enter a
parabolic orbit, (b) the decrease in speed required at point A for the
satellite to enter an elliptic orbit of minimum altitude 6370 km, (c) the
eccentricity
of the elliptic orbit.
PROBLEM 12.104 (Continued)
PROBLEM 12.105
A space probe is to be placed in a circular orbit of 5600 mi radius
about the planet Venus in a specified plane. As the probe reaches
A, the point of its original trajectory closest to Venus, it is inserted
in a first elliptic transfer orbit by reducing its speed by Δ.
A
v This
orbit brings it to Point B with a much reduced velocity. There the
probe is inserted in a second transfer orbit located in the specified
plane by changing the direction of its velocity and further reducing
its speed by Δ.
B
v Finally, as the probe reaches Point C, it is
inserted in the desired circular orbit by reducing its speed by Δ.
C
v
Knowing that the mass of Venus is 0.82 times the mass of the earth,
that 3
9.3 10 mi
A
r and 3
190 10
B
rmi, and that the probe
approaches A on a parabolic trajectory, determine by how much the
velocity of the probe should be reduced (a) at A, (b) at B, (c) at C.
PROBLEM 12.105 (Continued)
PROBLEM 12.106
For the space probe of Problem 12.105, it is known that rA3
9.3 10 mi and that the velocity of the probe is
reduced to 20,000 ft/s as it passes through A. Determine (a) the distance from the center of Venus to Point B,
(b) the amounts by which the velocity of the probe should be reduced at B and C, respectively.
PROBLEM 12.106 (Continued)
PROBLEM 12.107
As it describes an elliptic orbit about the sun, a spacecraft
reaches a maximum distance of 6
202 10 mi from the center
of the sun at Point A (called the aphelion) and a minimum
distance of 6
92 10mi at Point B (called the perihelion). To
place the spacecraft in a smaller elliptic orbit with aphelion at
A
and perihelion at ,
B
where
A
and
B
are located
6
164.5 10mi and 6
85.5 10mi, respectively, from the center
of the sun, the speed of the spacecraft is first reduced as it
passes through A and then is further reduced as it passes
through .B
Knowing that the mass of the sun is 332.8 3
10
times the mass of the earth, determine (a) the speed of the
spacecraft at A, (b) the amounts by which the speed of the
spacecraft should be reduced at A and
B
to insert it into the
desired elliptic orbit.