Planetary orbits

(4.81) | ||||||

(4.82) | ||||||

and | (4.83) |

where use has been made of Equation (4.75). It, thus, follows from Equations (4.79) and (4.80) that

According to Table 4.1, the planets also have low-eccentricity orbits, characterized by . In this situation, Equations (4.68)-(4.70) can be usefully solved via series expansion in to give

(See Section A.10.)

The preceding expressions can be combined with Equations (4.67), (4.77), (4.84), and (4.85) to produce

(4.88) | ||||||

(4.89) | ||||||

(4.90) | ||||||

(4.91) | ||||||

and | (4.92) |

Here, is expressed in radians per year, and in astronomical units. These equations, which are valid up to first order in small quantities (i.e., and ), illustrate how a planet's six orbital elements-- , , , , , and --can be used to determine its approximate position relative to the Sun as a function of time. The planet reaches its perihelion point when the mean ecliptic longitude, , becomes equal to the longitude of the perihelion, . Likewise, the planet reaches its aphelion point when . Furthermore, the ascending node corresponds to , and the point of furthest angular distance north of the ecliptic plane (at which ) corresponds to .

Consider the Earth's orbit about the Sun. As has already been mentioned, ecliptic longitude is measured relative to
a point on the ecliptic circle--the circular path that the Sun appears to trace out against the backdrop of the stars--known
as the vernal equinox. When the Sun reaches the vernal equinox, which it does every year on about March 20, day and
night are equally long everywhere on the Earth (because the Sun lies in the Earth's equatorial plane). Likewise, when the Sun reaches the opposite point on the ecliptic
circle, known as the *autumnal equinox*, which it does every year on about September 22, day and night are
again equally long everywhere on the Earth. The points on the ecliptic circle half way (in an angular sense) between the equinoxes are
known as the solstices. When the Sun reaches the *summer solstice*, which it does every year on about June 21,
this marks the longest day in the Earth's northern hemisphere, and the shortest day in the southern hemisphere.
Likewise, when the Sun reaches the *winter solstice*, which it does every year on about December 21, this
marks the shortest day in the Earth's northern hemisphere and the longest day in the southern hemisphere.
The period between (the Sun reaching) the vernal equinox and the summer solstice is known as *spring*, that
between the summer solstice and the autumnal equinox as *summer*, that between the autumnal equinox
and the winter solstice as *autumn*, and that between the winter solstice and the next vernal equinox as *winter*.

Let us calculate the approximate lengths of the seasons. It follows, from the preceding discussion, that the ecliptic longitudes of the Sun, relative to the Earth, at the (times at which the Sun reaches the) vernal equinox, summer solstice, autumnal equinox, and winter solstice are , , , and , respectively. Hence, the ecliptic longitudes, , of the Earth, relative to the Sun, at the same times are , , , and , respectively. Now, the mean longitude, , of the Earth increases uniformly in time at the rate of per year. Thus, the length of a given season is simply the fraction of a year, where is the change in mean longitude associated with the season. Equation (4.91) can be inverted to give

(4.93) |

to first order in . Hence, the mean longitudes associated with the autumnal equinox, winter solstice, vernal equinox, and summer solstice, are

(4.94) | ||||||

(4.95) | ||||||

(4.96) | ||||||

and | (4.97) |

respectively. (Recall that, according to Table 4.1, and for the Earth.) Thus,

(4.98) | ||||||

(4.99) | ||||||

(4.100) | ||||||

and | (4.101) |

(See Figure 4.7.) Given that the length of a