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- Derive Equations (5.2) and (5.7).

- Prove that in the case of a central force varying inversely as the cube of the
distance,

where
,
,
are constants. (From Lamb 1923.)

- The orbit of a particle moving in a central field is a circle that passes
through the origin; that is,
, where
. Show that the force law
is inverse fifth power. (Modified from Fowles and Cassiday 2005.)

- The orbit of a particle moving in a central field is the cardoid
, where
. Show that the force law
is inverse fourth power.

- A particle moving in a central field describes a spiral orbit
, where
,
.
Show that the force law is inverse cube, and that
varies logarithmically with
.
Demonstrate that there are two other possible types of orbit in this force field, and give their
equations. (Modified from Fowles and Cassiday 2005.)

- A particle moves in the spiral orbit
, where
. Suppose that
increases linearly
with
. Is the force acting on the particle central in nature? If not, determine how
would have to
vary with
in order to make the force central. Assuming that the force is central,
demonstrate that the particle's potential energy per unit mass is

where
is its (constant) angular momentum per unit mass.
(Modified from Fowles and Cassiday 2005.)

- A particle moves under the influence of a central force per unit mass of the form

where
and
are positive constants. Show that the associated orbit can be written

which is a closed ellipse for
and
. Discuss the character of the orbit for
and
.
Demonstrate that

where
.

- A particle moves in a circular orbit of radius
in an attractive
central force field of the form
, where
and
.
Demonstrate that the orbit is only stable provided that
.

- A particle moves in a circular orbit in an attractive
central force field of the form
, where
. Show
that the orbit is unstable to small perturbations.

- A particle moves in a nearly circular orbit of radius
under the action of the radial
force per unit mass

where
and
. Demonstrate that the so-called *apse line*, joining successive apse points, rotates
in the same direction as the orbital motion through an angle
each revolution. (From Lamb 1923.)

- A particle moves in a nearly circular orbit of radius
under the action of the central potential
per unit mass

where
and
. Show that the apse line rotates
in the same direction as the orbital motion through an angle
each revolution. (From Lamb 1923.)

- Suppose that the solar system were embedded in a tenuous uniform dust cloud. Demonstrate that the apsidal
angle of a planet in a nearly circular orbit around the Sun would be

where
is the mass of the Sun, and
is the mass of dust enclosed by a sphere whose radius matches the
major radius of the orbit. It is assumed that
.

- Consider a satellite orbiting around an idealized planet that takes the form of a uniform spheroidal mass
distribution of mean radius
and ellipticity
(where
). Suppose that the orbit
is nearly circular, with a major radius
, and lies in the equatorial plane of the planet. The potential
energy per unit mass of the satellite is thus (see Chapter 3)

where
is a radial coordinate in the equatorial plane. Demonstrate that the apse line rotates in the same
direction as the orbital motion at the rate

where
is the mean orbital angular velocity of the satellite.

** Next:** Rotating reference frames
** Up:** Orbits in central force
** Previous:** Perihelion precession of Mercury
Richard Fitzpatrick
2016-03-31