788 Encyclopedia of the Solar System
In this article the basic orbital properties of solar sys-
tem objects (planets, moons, minor bodies, and dust) and
their mutual interactions are described. Several examples
are provided of important dynamical processes that occur
in the solar system and groundwork is laid for describing
some of the phenomena that are discussed in more detail
in other articles of this encyclopedia.
1.1 Kepler’s Laws of Planetary Motion
By analyzing Tycho Brahe’s careful observations of the or-
bits of the planets, Johannes Kepler deduced the following
three laws of planetary motion:
1.All planets move along elliptical paths with the Sun
at one focus. The heliocentric distancer(i.e., the planet’s
distance from the Sun) can be expressed as
r=a(1−e^2 )
1 +ecosf, (1)withathe semimajor axis (average of the minimum and
maximum heliocentric distances) ande(the eccentricity
of the orbit)≡(1−b^2 /a^2 )^1 /^2 , where 2bis the minor axis
of an ellipse. The true anomaly, f, is the angle between
the planet’s perihelion (closest heliocentric distance) and
its instantaneous position (Fig. 1).
bar
fFIGURE 1 Geometry of an elliptical orbit. The Sun is at one
focus and the vectorrdenotes the instantaneous heliocentric
location of the planet (i.e.,ris the planet’s distance from the
Sun).ais the semimajor axis (average heliocentric distance),
andbis the semiminor axis of the ellipse. The true anomaly,f,
is the angle between the planet’s perihelion (closest heliocentric
distance) and its instantaneous position.
2.A line connecting a planet and the Sun sweeps out
equal areasAin equal periods of timet:A
t=constant. (2)Note that the value of this constant differs from one planet
to the next.
3.The square of a planet’s orbital periodPabout the
Sun (in years) is equal to the cube of its semimajor axisa
(in AU):P^2 =a^3. (3)1.2 EllipticaL Motion, Orbital Elements, and the
Orbit in Space
The Sun contains more than 99.8% of the mass of the known
solar system. The gravitational force exerted by a body is
proportional to its mass (Eq. 5), so to an excellent first ap-
proximation the motion of the planets and many other bod-
ies can be regarded as being solely due to the influence of
a fixed central pointlike mass. For objects like the planets,
which are bound to the Sun and hence cannot go arbitrar-
ily far from the central mass, the general solution for the
orbit is the ellipse described by Eq. (1). The orbital plane,
although fixed in space, can be arbitrarily oriented with re-
spect to whatever reference plane is chosen (such as Earth’s
orbital plane about the Sun, which is called theecliptic,or
the equator of the primary). The inclination,i, of the orbital
plane is the angle between the reference plane and the or-
bital plane and can range from 0 to 180◦. Conventionally,
bodies orbiting in a direct sense, with orbital angular mo-
mentum vectors within 90◦of the direction of the Earth’s
orbital angular momentum (or the rotational angular mo-
mentum of the primary), are defined to have inclinations
from 0◦to 90◦and are said to be on prograde orbits. Bod-
ies traveling in the opposite direction are defined to have
inclinations from 90◦to 180◦and are said to be on retro-
grade orbits. The two planes intersect in a line called the
line of nodes and the orbit pierces the reference plane at
two locations—one as the body passes upward through the
plane (the ascending node) and one as it descends (the de-
scending node). A fixed direction in the reference plane is
chosen and the angle to the direction of the orbit’s ascend-
ing node is called the longitude of the ascending node,.
Finally, the angle between the line to the ascending node
and the line to the direction of periapse (perihelion for or-
bits about the Sun, perigee for orbits about Earth) is called
the argument of periapseω. An additional angle, the lon-
gitude of periapseω=ω+is sometimes used in place
ofω. The six orbital elementsa,e,i,,ωandfuniquely
specify the location of the object in space (Fig. 2). The first