Fundamental Astronomy

Fig. 6.3. The radial velocity ˙r is the projection of the velocity
vector ˙r in the direction of the radius vector r
since k is a constant vector. Combining this with the
previous equation, we have
d
(k× ˙r + µr/r) = 0
dt
and
k× ˙r + µr/r = const =−µe . (6.10)
Since k is perpendicular to the orbital plane, k× ˙r
must lie in that plane. Thus, e is a linear combination
of two vectors in the orbital plane; so e itself must be
in the orbital plane (Fig. 6.4). Later we shall see that it
points to the direction where the planet is closest to the
Sun in its orbit. This point is called the perihelion.
One more constant is found by computing ˙r · ¨r:
˙r · ¨r =−µ˙r · r/r 3 =−µr˙r/r 3
=−µ˙r/r 2 = d
dt (µ/r).
Since we also have
˙r · ¨r = d
1
˙r · ˙r ,
dt 2
we get
d 1 µ
˙r · ˙r − = 0
dt 2 r
or
1
2 v2 − µ/r = const = h . (6.11)
Here v is the speed of the planet relative to the Sun. The
constant h is called the energy integral; the total energy
of the planet is m2h. We must not forget that energy
6.2 Solution of the Equation of Motion
Fig. 6.4. The orbit of an object in the gravitational field of another
object is a conic section: ellipse, parabola or hyperbola.
Vector e points to the direction of the pericentre, where the orbiting
object is closest to central body. If the central body is
the Sun, this direction is called the perihelion; if some other
star, periastron; if the Earth, perigee, etc. The true anomaly f
is measured from the pericentre
and angular momentum depend on the coordinate frame
used. Here we have used a heliocentric frame, which in
fact is in accelerated motion.
So far, we have found two constant vectors and one
constant scalar. It looks as though we already have seven
integrals, i. e. one too many. But not all of these constants
are independent; specifically, the following two
relations hold:
k· e = 0 , (6.12)
µ 2 (e 2 − 1) = 2hk 2 , (6.13)
where e and k are the lengths of e and k. The first
equation is obvious from the definitions of e and k. To
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