Using the planetary distance determination method that he derived in
conjunction with the positional observations of the planets that
Tycho Brahe had painstakingly accumulated over 20 years, Kepler
set about tracing the orbit of Mars. He soon found that the age-old idea
of circular orbits had to be discarded, as the distance of Mars from the
Sun appeared to vary. Kepler tested many models of orbital
shapes and finally showed that the orbital planes of the planets pass through
the Sun and that the orbital shape was an ellipse. These findings were
announced in 1609 as Kepler's first law - the law of ellipses:
The orbit of each planet is an ellipse with the Sun at one focus.
An ellipse is defined mathematically as the locus of all
points such that the sum of the distances from two foci to any point
on the ellipse is a constant. Figure 31
shows the elliptical orbit of a planet, P, with the
Sun, S, at one of the foci. The other focus, F, is
often called the empty focus. From the definition
of an ellipse, we have
r + r´ = 2a = constant.
figure 31:
An elliptical planetary orbit.
We define the following properties of the ellipse
in Figure 31:
major axis - the line AA´, where A
and A´ are the vertices of the ellipse.
semi-major axis - the lines CA and CA´,
where C is the centre of the ellipse. For each point P
on the ellipse at a distance r from focus S, there is
a symmetrical point P´ a distance r´ from
S - the average of these distances is
(r + r´) / 2 = a. This result holds for any
arbitrary but symmetrical pair of points. Hence the semi-major axis
is equal to the mean distance from the Sun of a planet in an elliptical orbit.
minor axis - the line BB´.
semi-minor axis - the lines CB and CB´.
If a and b denote the lengths of the semi-major and
semi-minor axes, respectively, then using the dashed lines in
Figure 31
(r = r´ = a)
and the Pythagoras theorem, we find
b^{2} = a^{2} -
a^{2}e^{2} =
a^{2}(1 - e^{2}),
where e is the eccentricity of ellipse, as described next.
eccentricity - the ratio CS / CA. The
eccentricity gives an idea of how elongated the ellipse is. If the ellipse
is a circle, e=0, since S and F are coincident
with C. The other limit for e is 1, obtained when the
ellipse is so narrow that the empty focus is removed to infinity.
The distance from each focus to the centre of the ellipse is ae.
perihelion - when the planet, P, is at A.
It is then nearest the Sun and we can write
SA = CA - CS = a - ae =
a (1 - e). If the gravitating body is not the Sun but some other body, then this point is known as the periapsis.
aphelion - when the planet, P, is at A´.
It is then farthest from the Sun and we can write
SA´ = CA´ + CS =
a + ae = a (1 + e).
If the gravitating body is not the Sun but some other body, then this point is
known as the apoapsis.
true anomaly - the angle ASP.
It is important to know the equation of an ellipse, as this can tell
us the distance from one focus to a point on the ellipse (e.g. the
Earth-Sun distance) as a function of the position of the point on
the ellipse. If we centre a polar coordinate system
(r, ) at S
and let the line SA correspond to
=0, then r measures
the distance SP, and
, the true anomaly, measures
the counter-clockwise angle ASP. Using
cos (180° - ) =
- cos
and the cosine law of plane trigonometry, we have
r´^{2} = r^{2} +
(2ae)^{2} +
2r(2ae)cos.
From the definition of an ellipse, we have
r´ = 2a - r
and so r = a (1 - e^{2}) / (1 + e cos
).
This is the equation of an ellipse in polar coordinates.
In Cartesian coordinates (x, y) the equation of the
ellipse can be derived using Figure 31 and
the theorem of Pythagoras, as follows:
r´^{2} = (x + ae)^{2} +
y^{2}
r^{2} = (x - ae)^{2} +
y^{2}
Subtracting these two equations and using the relation
r´ = 2a - r derived earlier, we find
r´ = a + ex. Substituting back into
the first of the above two equations and employing the relation
b^{2} = a^{2}(1 - e^{2}),
we obtain
(x/a)^{2} +
(y/b)^{2} = 1
which is the equation for an ellipse in Cartesian coordinates. It can
be seen that this equation reduces to the equation of a circle when
a=b.